Asymptotic Distribution and Simultaneous Confidence Bands for Ratios of Quantile Functions
AAsymptotic Distribution and Simultaneous ConfidenceBands for Ratios of Quantile Functions
Fabian Dunker Stephan Klasen Tatyana Krivobokova October 26, 2017
Abstract
Ratio of medians or other suitable quantiles of two distributions is widely used in med-ical research to compare treatment and control groups or in economics to comparevarious economic variables when repeated cross-sectional data are available. Inspiredby the so-called growth incidence curves introduced in poverty research, we arguethat the ratio of quantile functions is a more appropriate and informative tool tocompare two distributions. We present an estimator for the ratio of quantile func-tions and develop corresponding simultaneous confidence bands, which allow to assesssignificance of certain features of the quantile functions ratio. Derived simultaneousconfidence bands rely on the asymptotic distribution of the quantile functions ratioand do not require re-sampling techniques. The performance of the simultaneousconfidence bands is demonstrated in simulations. Analysis of the expenditure datafrom Uganda in years 1999, 2002 and 2005 illustrates the relevance of our approach.
Keywords and phrases : Growth incidence curve, Inequality, Quantile processes, Quantiletreatment effect, Pro-poor growth. School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140,New Zealand. Department of Economics, Georg-August-Universit¨at G¨ottingen, Platz der G¨ottinger Sieben 3, 37073G¨ottingen, Germany. Institute for Mathematical Stochastics, Georg-August-Universit¨at-G¨ottingen, Goldschmidtstr. 7,37077 G¨ottingen, Germany. a r X i v : . [ s t a t . M E ] O c t Introduction
Let X and X be two independent random variables with cumulative distribution func-tions F and F , respectively. The corresponding quantile functions are given by Q j ( p ) = F − j ( p ) = inf { x : F j ( x ) ≥ p } , j = 1 ,
2. In many applications it is of interest to comparequantiles of two random variables at a given p ∈ (0 , g ( p ) = Q ( p ) Q ( p ) . For example, if X is income in some population at time t and X is income at time t > t ,then g ( p ) reports the proportion by which the p -quantile of income changed from t to t ,with g ( p ) > g ( p ) shows the effect ofthe treatment on the p -quantile.In applications g ( p ) is either considered and interpreted at a fixed p ∈ (0 ,
1) or thecurve g ( p ), p ∈ (0 ,
1) is reduced to some number. For example, Cheng and Wu (2010)as well as Wu (2010) studied the effect of cancer treatment measured by the ratio of thecancer volumes in the treatment and the control group, the so-called
T /C -ratio. The
T /C -ratio can be formed for the mean cancer volume or for a certain quantile of the volumein the treatment and the control group, but typically is not considered as a function of p .Dominici et al. (2005) and Dominici and Zeger (2005) used the whole curve g ( p ), p ∈ (0 , E ( X ) − E ( X ) = (cid:90) { Q ( p ) − Q ( p ) } dp = (cid:90) [ Q ( p ) { − g ( p ) } ] dp which is known as the average treatment effect (ATE). To obtain ∆, log { g ( p ) } is estimatedby a smooth function. This approach has been applied to estimate the difference in medical2xpenditures between persons suffering from diseases attributable to smoking and personswithout these diseases.However, it is clearly more advantageous to view g ( p ) as a function of p . To the bestof our knowledge, this has been done only in the poverty research context. In particular,Ravallion and Chen (2003) used the curve G ( p ) = (cid:26) Q ( p ) Q ( p ) (cid:27) m − { g ( p ) } m − , p ∈ (0 , , m = 1 t − t ∈ (0 , t < t and called G ( p ) the growth incidence curve (GIC). Poverty reduction can be understood as increasingthe incomes of the poor. In this sense poverty is reduced from period t to t , if G ( p ) takespositive values for all small quantiles up the quantile where the poverty line was located inthe first period. Such growth that increases the incomes of poor quantiles has been called“weak absolute” pro-poor growth, i.e. growth that is accompanied by absolute povertyreduction without making any statement about the distributional pattern of growth, seeKlasen (2008). On the other hand, if G ( p ) has a negative slope, growth was pro-poorin the relative sense, i.e. the poor benefited (proportionately) more from growth thanthe non-poor. This means that such growth episodes led to a decrease in inequality andrelative poverty. For a detailed discussion of different notions of pro-poor growth we referto Ravallion (2004) and Klasen (2008). Growth incidence curves were also applied tonon-income data in Grosse et al. (2008).Hence, considering the whole curves g ( p ) or G ( p ), p ∈ (0 ,
1) provides more informativecomparison of two distributions and can be applied not only in the poverty research context.The goal of this work is to derive the asymptotic distribution of an estimator of g ( p ) andbuild simultaneous confidence bands for g ( p ). Estimation and inference for G ( p ) is thenstraightforward. 3ominici et al. (2005) proposed an estimator for log { g ( p ) } using smoothing splines.Venturini et al. (2015) extend the work by Dominici et al. (2005), employing a Bayesianapproach to get a smooth estimator of h { g ( p ) } , for some known monotone differentiablefunction h . A much simpler approach, which we pursue, would be to replace the unknown Q j ( p ) in g ( p ) by some estimator (cid:98) Q j ( p ), j = 1 , (cid:98) g ( p ). There are several quantileestimators available (see e.g. Harrell and Davis, 1982; Kaigh and Lachenbruch, 1982; Cheng,1985). In this work we employ the classical empirical quantile function.Apparently, simultaneous inference about the curve g ( p ), p ∈ (0 ,
1) is crucial in ap-plications, but has not been considered so far, to the best of our knowledge. Dominiciet al. (2005) rather focused on estimation of the average treatment effect with the help oflog { g ( p ) } and do not discuss inference about g ( p ). Cheng and Wu (2010) consider estima-tion of g ( p ) at a given p ∈ (0 ,
1) and build a confidence interval for g ( p ) using asymptoticnormality arguments and several estimators for the variance of (cid:98) g ( p ). The WorldbankPoverty Analysis Toolkit (can be found at http://go.worldbank.org/YF9PVNXJY0) pro-vides also only point-wise confidence intervals for growth incidence curves, similar in spiritto that of Cheng and Wu (2010). More specifically, the confidence statement in this toolkitis constructed for a discretization of (0 ,
1) by 0 < p < p < . . . < p k <
1. For every p i , i = 1 , . . . , k expectation and variance for some estimator (cid:98) G ( p i ) of G ( p i ) are estimated with abootstrap. Critical values c i and c i are then taken from the corresponding t -distribution forsome level α . This implicitly assumes that (cid:98) G ( p i ) is asymptotically normal. The resultingconfidence statement has the form P { c i ≤ G ( p i ) ≤ c i } = 1 − α, for each i = 1 , , . . . , k, where α ∈ (0 ,
1) is some pre-specified confidence level. Obviously, these confidence intervalsprovide inference only at a given p i . For example, if we would like to test significance of the4overty reduction (or treatment effect) at the median, it is enough to build a point-wiseconfidence interval for G (0 .
5) = { g (0 . } m − g (0 . G ( p ) has to be made and, hence, simultaneous confidencebands should be considered. That is, the goal is to find such c ( p ) and c ( p ) that P { c ( p ) ≤ G ( p ) ≤ c ( p ) for all p ∈ (0 , } = 1 − α. The difference to the point-wise intervals is that c ( p ) ≤ G ( p ) ≤ c ( p ) holds not only sepa-rately for every p , but simultaneously for all p ∈ (0 , g ( p ) or G ( p ), the analysis of the asymptoticdistribution of the function (cid:98) g ( p ) is necessary. This involves the theory of empirical processeswhich goes back to Glivenko (1933), Cantelli (1933), Donsker (1952), and Koml´os et al.(1975). Our analysis builds on results for empirical quantile processes and its simultaneousconfidence bands developed in Cs¨org˝o and R´ev´esz (1978), Cs¨org˝o and R´ev´esz (1984), andCs¨org˝o (1983). The main benefit of this approach is that it allows for faster computationof the confidence bands without re-sampling techniques.The paper is organized as follows. In Section 2 we introduce a simple sample counter-part estimator and analyse its asymptotic distribution. This estimator is also used by theWorld Bank Toolkit. The results about the asymptotic distribution motivates two con-structions for asymptotic simultaneous confidence bands presented in Section 3. Section 4evaluates the small sample properties of our confidence bands by Monte Carlo simulations.Expenditure data from Uganda are analysed with our confidence bands in Section 5 beforewe conclude in Section 6. 5 Estimation and asymptotic distribution
Throughout this section we assume that we have i.i.d. samples X , , X , . . . X ,n of X and X , , X , . . . X ,n of X . Furthermore, we assume that the samples are stochasticallyindependent of each other. This assumption is justified if the data are collected in twoindependent groups (e.g. treatment and control) or in repeated cross-sections. Note thatthere is a related concept of non-anonymous growth incidence curves proposed for paneldata in Grimm (2007) and Bourguignon (2011). Non-anonymous growth incidence curvesare built based on two dependent samples and are not treated in this work. We start by presenting a simple sample estimator for g ( p ) and G ( p ). For j = 1 , k -th order statistic of the sample X j, , X j, . . . X j,n j by X j, ( k ) . The sample quantilefunction is the inverse of the right continuous empirical distribution function, which isknown to be (cid:98) Q j ( p ) = (cid:98) F − j ( p ) = X j, ( k ) , for k − n j < p ≤ kn j , k = 1 , , . . . , n j , j = 1 , . (1)We now define estimators of g ( p ) and G ( p ) as (cid:98) g ( p ) = (cid:98) Q ( p ) (cid:98) Q ( p ) and (cid:98) G ( y ) = { (cid:98) g ( p ) } m − , m ∈ (0 , . (2)It is well-known that the quantile function and its empirical version are equivariant un-der strictly monotone transformations. Let us denote by F j and Q j = F − j the cumulativedistribution and quantile functions of X j = log( X j ), j = 1 ,
2, respectively. Also, let (cid:98) Q j bethe empirical quantile function of the log-transformed sample X j,i = log( X j,i ), i = 1 , . . . , n j ,6 = 1 ,
2. Then, Q j = log( Q j ), as well as (cid:98) Q j = log( (cid:98) Q j ), j = 1 ,
2. Consequently,log { g ( p ) } = Q ( p ) − Q ( p ) , log { (cid:98) g ( p ) } = (cid:98) Q ( p ) − (cid:98) Q ( p )log { G ( p ) + 1 } = m {Q ( p ) − Q ( p ) } , log { (cid:98) G ( p ) + 1 } = m { (cid:98) Q ( p ) − (cid:98) Q ( p ) } . (3)Hence, a simultaneous confidence band for g ( p ) can be obtained observing that P { c ( p ) ≤ g ( p ) ≤ c ( p ) , ∀ p ∈ (0 , } = P [log { c ( p ) } ≤ Q ( p ) − Q ( p ) ≤ log { c ( p ) } , ∀ p ∈ (0 , . Note that the difference of two quantile functions ∆( p ) = Q ( p ) − Q ( p ) is known as quantiletreatment effect (QTE), sometimes also named the percentile-specific effect between twopopulations, see Dominici et al. (2006). To the best of our knowledge, the inference forQTE is usually done at a fixed p ∈ (0 , We first characterizes the asymptotic distribution of (cid:98) G ( p ) at a fixed p ∈ (0 , Assumption 1.
Two independent random variables X > a.s. and X > a.s. withfinite second moments and cumulative distribution functions F and F are given togetherwith random samples X j, , X j, , . . . , X j,n j , j = 1 ,
2. The log-transformed X j = log( X j ) hasthe cumulative distribution function F j and density f j = F (cid:48) j , j = 1 ,
2. The correspondingquantile function Q j ( p ) = F − j ( p ) has the quantile density q j ( p ) = Q (cid:48) ( p ) = 1 /f j {Q j ( p ) } , p ∈ (0 , j = 1 , Theorem 1.
Let Assumption 1 hold and p ∈ (0 , be fixed. Moreover, assume F and F are continuously differentiable at some x and x , respectively, such that F ( x ) = F ( x ) = p and f ( x ) > , f ( x ) > . i) For min { n , n } → ∞ the estimator (cid:98) G ( p ) + 1 = { (cid:98) g ( p ) } m is asymptotically log-normalwith the parameters µ ( p ) = m log { g ( p ) } and σ ( p ) = (cid:115) m p (1 − p ) (cid:20) { q ( p ) } n + { q ( p ) } n (cid:21) . (ii) If in addition F and F are continuously differentiable at some ˜ x and ˜ x , respec-tively, such that F (˜ x ) = F (˜ x ) = ˜ p , for some < p ≤ ˜ p < , and f (˜ x ) > , f (˜ x ) > , then the asymptotic distribution of { (cid:98) G ( p ) + 1 , (cid:98) G (˜ p ) + 1 } is bivariatelog-normal with the parameters { µ ( p ) , µ (˜ p ) } and σ ( p, ˜ p ) = m p (1 − ˜ p ) (cid:26) q ( p ) q (˜ p ) n + q ( p ) q (˜ p ) n (cid:27) . Corollary 1.
Under the assumptions of Theorem 1 we have asymptotic normality for (cid:98) G ( p ) + 1 = { (cid:98) g ( p ) } m in the sense that (cid:98) G ( p ) + 1 − { g ( p ) } m { g ( p ) } m σ ( p ) D −→ N (0 , converges in distribution to a standard normal random variable for min { n , n } → ∞ andfor any fixed p ∈ (0 , . The World Bank Toolkit and Cheng and Wu (2010) implicitly employ the asymptoticnormality of (cid:98) G ( p ) and (cid:98) g ( p ) to build point-wise confidence intervals, but use different vari-ance estimators, based either on bootstrap or on certain approximations. To the best ofour knowledge, the result of Corollary 1 is new. Note also that σ ( p ) depends on unknown q j ( p ), j = 1 ,
2, which have to be consistently estimated in practice.Theorem 1 and Corollary 1 provide two different ways for deriving point-wise confi-dence statements about G ( p ) (or about g ( p ) by setting m = 1). We can approximate the8istribution of (cid:98) G ( p ) + 1 = { (cid:98) g ( p ) } m for a fixed p ∈ (0 ,
1) either by a log-normal or bya normal distribution. However, the log-normal approximation is preferable for positiverandom variables. Indeed, X j > a.s. , j = 1 , g ( p ) ∈ [0 , ∞ ) for all p ∈ (0 , (cid:98) G ( p ) + 1 = { (cid:98) g ( p ) } m puts probabilitymass outside of [0 , ∞ ). This can cause confidence intervals to take impossible values, inparticular in small samples, and affect the actual coverage of the band. Taking a log-normalapproximation helps to avoid this. We use the log-normal approximation implicitly in ourconstructions of simultaneous confidence bands in Section 3. In the previous Section 2.2 derivation of the confidence statements about G ( p ) or g ( p ) at oneor at a finite number of points reduces to finding the limiting distribution of (cid:98) Q ( p ) − (cid:98) Q ( p )at a fixed p ∈ (0 , G ( p ) or g ( p ) that hold for all p ∈ (0 ,
1) simultaneously, we need to find the limiting distribution of (cid:98) Q ( p ) − (cid:98) Q ( p ), whichis treated as a stochastic process indexed in p ∈ (0 , D n ,n ( p ; s ) = (cid:114) n n n + s n (cid:40) s (cid:98) Q ( p ) − Q ( p ) q ( p ) − (cid:98) Q ( p ) − Q ( p ) q ( p ) (cid:41) , p ∈ (0 , , where s > n needed later for technical reasons.For the analysis of this process we need the following set of assumptions on X and X . Assumption 2.
The cumulative distribution functions F j of the log-transformed X j =log( X j ), j = 1 , a, b ), where a = sup { x : F j ( x ) = 0 } , b =inf { x : F j ( x ) = 1 } , −∞ ≤ a < b ≤ ∞ and f j > a, b ). In addition, there exists some9 < γ < ∞ such that sup x ∈ ( a,b ) F j ( x ) { − F j ( x ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) j ( x ) { f j ( x ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ γ, j = 1 , . (4) Assumption 3.
For A j = lim sup x (cid:38) a f j ( x ) ≤ ∞ and B j = lim sup x (cid:37) b f j ( x ) ≤ ∞ , j = 1 , A j , B j ) > A j = B j = 0, then f j is non-decreasing on an interval to the right of a andnon-increasing on an interval to the left of b .If X and X are log-normal, as typically the case for income, expenditure and similarpositive random variables, then f j is the density of a normal distribution. Hence, existence,positivity and differentiability of f j on R are trivially fulfilled. The supremum in (4) is 1for normally distributed random variables independent of expectation and variance. Theproperty in Assumption 3 is called tail-monotonicity. For normal distributions A j = B j = 0and Assumption 3 (ii) obviously holds.The following result shows that D n ,n ( p ; s ) converges uniformly to a Brownian bridge B ( p ). Recall that a Brownian bridge is a standard Wiener process W ( p ) with W (0) = W (1) = 0, i.e. B ( p ) = W ( p ) − pW (1), p ∈ [0 , B ( p ) ∼ N (0 , p − p ) and Cov { B ( p ) , B (˜ p ) } = p (1 − ˜ p ) for all 0 ≤ p ≤ ˜ p ≤ Theorem 2.
Let Assumptions 1 and 2 hold and set n = min { n , n } . Then a series ofBrownian bridges B n ,n can be defined such that for any fixed s sup p ∈ [ δ n , − δ n ] (cid:12)(cid:12)(cid:12) D n ,n ( p ; s ) − B n ,n ( p ) (cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) with δ n = 25 n − log log( n ) . If in addition Assumption 3 holds, a Brownian bridge B n ,n an be defined such that in case of Assumption 3 (i) sup p ∈ [0 , (cid:12)(cid:12)(cid:12) D n ,n ( p ; s ) − B n ,n ( p ) (cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) and in case of Assumption 3 (ii) sup p ∈ (0 , (cid:12)(cid:12)(cid:12) D n ,n ( p ; s ) − B n ,n ( p ) (cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) if γ < (cid:2) n − / { log log( n ) } γ { log( n ) } (1+ ε )( γ − (cid:3) if γ ≥ for arbitrary ε > . For example, if X j are approximately log-normal in a way that log( X j ) has the tail be-havior of a normal variable, then according to Theorem 2 the process D n ,n ( p ; s ) convergesto a Brownian bridge simultaneously on (0 ,
1) with the rate O { n − / log( n ) } .Constructing confidence sets for g ( p ) or G ( p ) = { g ( p ) } m − (cid:98) Q ( p ) − (cid:98) Q ( p ) = log { (cid:98) g ( p ) } = m − log { (cid:98) G ( p ) + 1 } , while D n ,n ( p ; s ) in Theorem 2 contains s (cid:98) Q ( p ) /q ( p ) − (cid:98) Q ( p ) /q ( p ) instead. Therefore, let usconsider D ∗ n ,n ( p ; s ) = 2 (cid:114) n n n + s n (cid:98) Q ( p ) − Q ( p ) − (cid:110) (cid:98) Q ( p ) − Q ( p ) (cid:111) q ( p ) /s + q ( p ) . and discuss the choice of s . First, introduce the following assumption. Assumption 4.
There exists a constant s > q ( p ) = sq ( p ), p ∈ (0 , D ∗ n ,n ( p ; s ) = D n ,n ( p ; s ) = (cid:114) n n n + s n (cid:98) Q ( p ) − Q ( p ) − (cid:110) (cid:98) Q ( p ) − Q ( p ) (cid:111) q ( p )11nd Theorem 2 can be applied to get the asymptotic distribution of (cid:98) Q ( p ) − (cid:98) Q ( p ) andhence the simultaneous confidence bands for G ( p ) or g ( p ).It is shown in the Appendix, that if Assumption 4 is true, then s = (cid:82) ∞−∞ { f ( x ) } dx (cid:82) ∞−∞ { f ( x ) } dx . (5)Moreover, if the X j have distribution from the location-scale family of distributionswith locations µ j and scales σ j < ∞ , j = 1 ,
2, then Assumption 4 implies that s ∝ σ /σ .This can be seen directly from (5) applying the change of variable y = µ j + σ j x . Also,let (cid:101) Q j denote the quantile function of {X j − µ j } /σ j and ˜ q j the corresponding quantiledensity. Then, Q j ( p ) = µ j + σ j (cid:101) Q j ( p ) and therefore q j ( p ) = σ j ˜ q j ( p ), p ∈ (0 , j = 1 , q ∝ ˜ q and thus the distributions of X and X differ only in location and scale parameters.For example, if X j are both log-normally distributed with arbitrary location parametersand scale parameters σ j , then log( X j ) = X j , j = 1 , s = σ /σ . In applications, to check if distributions of X and X differ only in the location andscale, one can inspect the QQ-plot of standardised log-transformed data.If the quantile densities are not proportional, that is, Assumption 4 is not fulfilled, wehave to handle the term D ∗ n ,n ( p ; s ) − D n ,n ( p ; s ) = q ( p ) − s q ( p ) q ( p ) + s q ( p ) (cid:114) n n n + s n (cid:40) (cid:98) Q ( p ) − Q ( p ) q ( p ) /s + (cid:98) Q ( p ) − Q ( p ) q ( p ) (cid:41) . Lemma 1.
Under Assumptions 1, 2 and 3 lim sup n ,n →∞ (cid:18) log log (cid:114) n n n + s n (cid:19) − / sup p ∈ (1 /n, − /n ) (cid:12)(cid:12) D ∗ n ,n ( p ; s ) − D n ,n ( p ; s ) (cid:12)(cid:12) a.s. ≤ ν √ p ∈ (1 /n, − /n ) (cid:12)(cid:12)(cid:12)(cid:12) q ( p ) − s q ( p ) q ( p ) + s q ( p ) { p (1 − p ) } ν (cid:12)(cid:12)(cid:12)(cid:12) or all ν ∈ [0 , / . Note that the bound on the right hand side is always smaller are equal 1 / √ ν ∈ [0 , / q and q are usually similar functions in applications, much smallerbounds can be expected. Based on the results of the previous section, we can derive simultaneous confidence bandsfor Q ( p ) − Q ( p ) = log { g ( p ) } = m − log { G ( p ) + 1 } and transform them into simultaneousconfidence bands for g ( p ) or G ( p ). Note that simultaneous confidence bands for the quantiletreatment effect Q ( p ) − Q ( p ) follow immediately. We make use of Theorem 2 and Lemma1 from the last section, as well as the Kolmogorov distribution P (cid:32) sup p ∈ [0 , | B ( p ) | ≤ c (cid:33) = ∞ (cid:88) k = −∞ ( − k e − k c . (6)Throughout this section we assume a confidence level α and denote the correspondingcritical value for the Brownian bridge by c α such that P (cid:0) sup p ∈ [0 , | B ( p ) | ≤ c α (cid:1) = 1 − α . Inaddition, we denote by c s an asymptotically almost sure upper bound from Lemma 1 c s = inf ≤ ν ≤ / − δ (cid:18) log log (cid:114) n n n + s n (cid:19) / ν √ p ∈ (1 /n, − /n ) (cid:12)(cid:12)(cid:12)(cid:12) q ( p ) − s q ( p ) q ( p ) + s q ( p ) [ p (1 − p )] ν (cid:12)(cid:12)(cid:12)(cid:12) . with some δ > Q ( p ) − Q ( p ). Similar approachesfor the quantile function have been explored in Cs¨org˝o and R´ev´esz (1984).13 .1 Confidence bands with quantile density estimation The first approach to the construction of confidence bands relies on the following argument1 − α ≈ P ( | D n ,n ( p ; s ) | ≤ c α , for all 0 < p < ≤ P (cid:0)(cid:12)(cid:12) D ∗ n ,n ( p ; s ) (cid:12)(cid:12) ≤ c α + c s , for all 0 < p < (cid:1) = P (cid:34) (cid:12)(cid:12)(cid:12) Q ( p ) − Q ( p ) − (cid:110) (cid:98) Q ( p ) − (cid:98) Q ( p ) (cid:111)(cid:12)(cid:12)(cid:12) ≤ ( c α + c s ) (cid:115) n + s n n n q ( p ) /s + q ( p )2 , for all 0 < p < (cid:35) . The quantities q j ( p ), j = 1 , q j ( p ) have been proposed, typically based on kerneldensity estimation, see e.g. Cs¨org˝o et al. (1991), Jones (1992), Cheng (1995), Cheng andParzen (1997), Soni et al. (2012), and Chesneau et al. (2016). We make the followingassumption on the densities. Assumption 5.
The densities f j , j = 1 , x ∈ ( a,b ) [ F j ( x ) { − F j ( x ) } ] f j ( x ) < ∞ and sup x ∈ ( a,b ) (cid:12)(cid:12)(cid:12) f (cid:48)(cid:48) j ( x ) (cid:12)(cid:12)(cid:12) < ∞ . Now we can get the simultaneous confidence bands for the difference of two quantilefunctions.
Theorem 3.
Let Assumptions 1, 2, 3 and 5 hold and let K be a second order kernel withsupport in [ − / , / . For j = 1 , set (cid:98) q j ( p ) = h − n j (cid:90) K (cid:18) y − zh n j (cid:19) d (cid:98) Q j ( z ) . hen a series of Brownian bridges B n ,n can be defined such that for any fixed s sup p ∈ [ ε n , − ε n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:114) n n n + s n (cid:40) (cid:98) Q ( p ) − Q ( p ) (cid:98) q ( p ) /s − (cid:98) Q ( p ) − Q ( p ) (cid:98) q ( p ) (cid:41) − B n ,n ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = ø (cid:40) (cid:112) log log( n ) n δ (cid:41) and for c ∗ α ( p ) = ( c α + c s ) (cid:115) n + s n n n (cid:98) q ( p ) /s + (cid:98) q ( p )2 (7) we get − α ≤ lim n ,n →∞ P (cid:110) (cid:98) Q ( p ) − (cid:98) Q ( p ) − c ∗ α ( p ) ≤ Q ( p ) − Q ( p ) (8) ≤ (cid:98) Q ( p ) − (cid:98) Q ( p ) + c ∗ α ( p ) , p ∈ ( ε n , − ε n ) (cid:111) with h n j = n − ηj , n = min { n , n } , ε n = n − β , β + δ < η < / , and η/ δ + 2 β < / . Note that if Assumption 4 holds, then c s in (7) is set to zero and s is chosen as in (5).Simultaneous confidence bands (8) are given for the difference of two quantile functions,known as the quantile treatment effect. To get simultaneous confidence bands for g ( p ) and G ( p ) recall that Q ( p ) − Q ( p ) = log { g ( p ) } = m − log { G ( p ) + 1 } so that P (cid:110) (cid:98) Q ( p ) − (cid:98) Q ( p ) − c ∗ α ( p ) ≤ Q ( p ) − Q ( p ) ≤ (cid:98) Q ( p ) − (cid:98) Q ( p ) + c ∗ α ( p ) , p ∈ ( ε n , − ε n ) (cid:111) = P { exp( − c ∗ α ( p )) (cid:98) g ( p ) ≤ g ( p ) ≤ exp( c ∗ α ( p )) (cid:98) g ( p ) , p ∈ ( ε n , − ε n ) } = P (cid:104)(cid:110) (cid:98) G ( p ) + 1 (cid:111) exp( − c ∗ α ( p ) m ) − ≤ G ( p ) ≤ (cid:110) (cid:98) G ( p ) + 1 (cid:111) exp( c ∗ α ( p ) m ) − , p ∈ ( ε n , − ε n ) (cid:105) . The confidence band above depends on nonparametric estimation of quantile densities.Two smoothing parameters h n j , j = 1 , Theorem 4.
Let Assumption 1 and 2 hold. Then − α = lim n ,n →∞ P (cid:40) (cid:98) Q (cid:18) p − c α √ n (cid:19) − (cid:98) Q (cid:18) p + c α √ n (cid:19) ≤ Q ( p ) − Q ( p ) ≤ (cid:98) Q (cid:18) p + c α √ n (cid:19) − (cid:98) Q (cid:18) p − c α √ n (cid:19) ; ε n ≤ y ≤ − ε n (cid:41) , with ε n = n − / δ for any δ ∈ (0 , / . Theorem 4 requires fewer assumptions than Theorem 3, but there is no explicit con-vergence rate given. However, these confidence bands give good results in numerical sim-ulations. To obtain simultaneous confidence bands for g ( p ) or G ( p ) use Q ( p ) − Q ( p ) =log { g ( p ) } = m − log { G ( p ) + 1 } . We evaluate the properties of the confidence bands by using synthetic data and buildingconfidence bands for growth incidence curves G ( p ). Confidence bands for the quantiletreatment effect and g ( p ) are equivalent. We consider two settings and in both of them fix m = 1. In the first setting X and X are drawn from log-normal distributions. Thereby, X has location parameter 0 and scale parameter σ = 0 .
7, while X has location parameter0 . σ = 1. As already discussed, Assumption 4 holds in this examplewith s = σ /σ = 0 .
7. This value is estimated in the simulations, while c s is set to zero.In the second setting, X is as in the first setting, while X is drawn from the gammadistribution with the shape parameter 2 and scale parameter 1. In this setting Assumption4 does not hold and c s is estimated for the plug-in confidence bands.16e considered four sample sizes n = n = n ∈ { , , ,
10 000 } . For proba-bility values p ∈ (0 ,
1) we used an equidistant grid of length 100 to build the confidencebands; setting the grid length to n does not change the results significantly, but increasesthe computation time in Monte Carlo simulations. The results are based on the MonteCarlo samples of size 5 000. The following Table 1 summarizes the actual coverage prob-ability with simulated data for 1 − α = 0 .
95. The results are given in both settings forthe confidence bands with plug-in estimators, for the direct confidence bands and for theconfidence bands built with the World Bank algorithm.Setting 1 Setting 2Sample size n Plug-in Direct World Bank Plug-in Direct World Bank100 0.888 0.965 0.460 0.893 0.964 0.3861 000 0.975 0.960 0.286 0.958 0.960 0.1775 000 0.980 0.959 0.343 0.969 0.960 0.26710 000 0.984 0.960 0.425 0.973 0.964 0.390Table 1: Coverage probability of the plug-in, direct and World Bank confidence bands.First of all, the coverage of the confidence bands obtained with the World Bank al-gorithm is way too small. The reason is that we tested simultaneous coverage, while theWorld Bank algorithm constructs only point-wise confidence bands.The actual coverage probability of all our constructions (about 0 .
96) is slightly largerthan the theoretical probability 0 .
95, except for the plug-in confidence bands for n = 100,where the coverage is lower than the nominal. This can be attributed to the quality of thenonparametric estimates of the quantile densities in small samples, as also expected fromTheorem 3. Once the sample size is large, both confidence bands perform very similar,even with the estimated correction c s for the plug-in bands in the second setting.The plots in Figure 1 show typical estimates from the first setting together with 95%plug-in and direct confidence bands for n = 100 (left) and n = 1 000 (right). The true17 .2 0.4 0.6 0.8 Probability G I C Probability G I C Figure 1: Estimates for growth incidence curves and 95% simultaneous confidence bandsfor n = 100 (left) and n = 1000 (right). Each plots shows the ture growth incidence curve(dahsed), its estimator (bold), plug-in confidence bands (grey area) and direct confidencebands (bold).growth incidence curve G ( p ) is the dashed line, while its estimate is the solid line. Plug-inconfidence bands are shown as a grey area, while direct confidence bands are solid linesenveloping the growth incidence curve. In accordance with the simulation results, plug-inconfidence bands are somewhat narrower for small n = 100, while for n = 1 000 bothconfidence bands are nearly indistinguishable. As stated in Theorem 3 and Theorem 4 theconfidence bands are not defined for p close to 0 and close to 1. The plots show the bandsfor probabilities p between ε n and 1 − ε n . Our work is motivated by the application of growth incidence curves to the evaluation ofpro-poorness of growth in developing countries. Absolute poverty is reduced if the growthincidence curve G ( p ) is positive for all income quantiles below the poverty line and such18rowth is called pro-poor using the weak absolute definition mentioned in the introduction.In this case, there is some income growth for the poor and absolute poverty is reduced. Inaddition, relative poverty is reduced if G ( p ) has a negative slope, such growth is called pro-poor using the relative definition as it is associated with declining inequality and decliningrelative poverty.We analyse data from the Uganda National Household Survey for the years 1992, 2002,and 2005. This is a standard multi-purpose household survey that is regularly conductedto monitor trends in poverty and inequality and its most important correlates. The samplesizes are n = 9923, n = 9710, and n = 7421. We measure welfare by householdexpenditure per adult equivalent in 2005 / (cid:98) s according to (5) and set c s = 0. l ll l ll ll l llll lll l l llll l ll lll lll lll ll l lll l lllllll ll lll ll ll lll ll lll ll ll l ll llll l ll l ll l ll lll ll llllll ll lllll l ll ll lll l llll ll l l llll ll ll lll lll lll l l ll lll lll l l lll l lllll l lll ll lll l ll lll ll llll l l llll ll lllll ll l llll llll lll lll ll llll lll l lll l ll ll llll llll ll ll l lllll ll ll ll l ll ll l llll ll llll l ll lll llll ll l lll l l l lllll ll lll ll ll ll l ll lll lll l lll ll ll lll l ll ll ll lllll ll lll ll lll l ll ll l lllllll l lll l lll lll lllll ll l lll l ll lll l lllll llllll ll ll l ll lll ll ll ll l l ll ll ll ll l ll lll lll l ll llll ll lll l ll ll ll lll l lllll lllll lll ll ll ll ll l ll ll ll lllll lll ll lll l ll l ll l ll ll ll ll ll l ll l ll l l lllll llll lll ll l lll lll ll llll l ll llllll lll l ll ll lllll ll ll l l llll ll l ll l l ll l ll ll ll ll llll lll l l ll ll lll ll ll ll l ll lll l ll lll lll lll lll l ll ll l ll lll ll l ll l ll llll lllll l ll ll l lllllll lll l ll l lll lll l llll ll l llll ll ll ll lll ll ll lll l lll llll llll ll ll lll lll ll l lllll ll lll l ll ll l llll lll ll l llll l lllll l llll ll l lll lll lll ll l lllll lll lll lll l llll l ll ll lll ll lll ll ll ll l ll l ll lll ll l lll l llll ll l ll ll ll l ll l l lll ll lll ll ll ll ll lllll l ll lll llll ll l lllll ll l ll ll l lll l ll ll llll lll lll ll l ll llllll ll l lll llll l ll l llll l ll l ll ll lll l lll l l ll ll ll lllll l l l ll llll llll l ll ll ll ll lll l lll l lll l l ll ll lll lll ll ll llll l ll lll l lll llll ll ll llll ll lll lll llll l lllll l lll ll l ll l ll llll lll l l ll ll ll lll l llll lll lll lll lll lll ll l ll lll ll llll ll ll lll lll ll lll lllll l lll l lllll llll l llll l ll l ll ll l ll lll ll ll l ll llll lll lll lll ll ll ll l ll l ll ll llll ll l lll lll llll ll l lll l lll lll l lll ll ll l ll l llll l l lll lll lll lllll ll ll lll l llll lll llll lll lll l ll lll ll l l lllll lllll l ll llll ll ll l lll ll ll l lll llll l lll lll llll l lll lll ll ll llll ll lll ll llll lll l lll l llll lll lll lll lll lll lll l ll ll lll l ll ll lll lll l l ll llll ll ll ll l lll ll lllll l ll llll ll ll lllll llll lll lll lll l llll l ll lll lll l ll l ll lll ll l lll ll llll l llll l ll l lll lll llll lll ll lll ll ll lll lll ll ll ll ll ll l lll l ll ll lll lll l l lll llll ll llll lll ll ll ll ll l ll ll lll l ll ll lll ll l ll lllll llll l lll ll llll ll ll l lll l ll ll ll l lll l l l ll lll ll l ll lll l ll l llll ll l lllll l ll lll llll ll l lll l lll l ll ll lll lll ll l lll l ll ll l lll l l lll lll lll lll ll ll ll l ll ll l llll llll ll ll llll ll lll llll llll l lllll llll l lll ll l ll ll l ll lll llll llll ll l ll l lll l ll lll ll lll l l l l ll ll llll lll ll ll ll ll ll lllll l lllll ll llll l l lll ll lll lll llll lllll l lll lllll lll ll lllll ll l lll l lll lll lll l l lll llll ll ll l l ll ll lll lll l ll llllll lll l ll lll l ll ll ll ll llll l lll ll ll ll lll ll l lll l l ll ll lll ll ll ll l llll lll ll ll l l lll l lll ll ll ll l lllll lll ll ll ll ll ll lll l lll lll lll ll ll lll llll ll ll l ll ll lll ll l ll ll l ll l llll lll l ll l l lllll l lll ll ll ll lll ll l lllll l ll ll lll ll lllllll lllll l llll l l ll l lll ll ll lll lll ll l l lll ll llll lll ll l ll l ll lll ll ll ll l lll lll l l llllll ll ll ll llll ll ll llll l ll lll llll lllll ll ll llll ll ll lll l ll ll llll l l l ll lll l llll ll l l lll l ll lll llll llll lll l lll lll l lll l lllll l ll ll llll ll ll lll llll l l lll l lll l ll l l ll l ll ll l ll l llll l lll llll l ll ll ll ll ll lll l llll ll l l lll ll ll ll ll ll l ll ll llll l ll llll ll ll ll ll llll ll ll l lll llll l lll ll ll l ll lll l ll lll ll ll l ll l ll lll l llll lll l l ll llll lll ll l llllll lll ll l lll l lllll l ll l ll lll llll l llll ll llll l lllll lllll ll l lll lll lll ll ll l ll ll ll ll ll ll l l llll lll l lll ll ll lll lll l lll l ll lllll ll lll ll lll lll ll llll llll ll lll l lll l llll lll l lllll ll llll lll lll lll ll lll l lllll l lll llll l ll lll l l ll ll l ll ll ll llll l lll llll lll l l ll l ll l llll lll ll ll l lll lll lll l ll ll l l lll l llll ll llll ll l ll l l lllll ll ll ll llll ll lll ll ll lll ll ll l llll ll ll ll l l l llllll ll llll ll lll lll llll ll ll l ll ll ll ll lll l llll ll l llll l lll l lll lll l llll ll ll ll ll ll l lll l ll l lll lll lll llll l l lll lll llll l l lll l llll lll ll l ll l l ll l llllll l lll ll llll llll ll llll l llllll ll l lll l ll ll llll l lll l ll lll l ll ll ll ll ll lll lll l llll lll ll lll ll llll ll llll ll lll l ll lll ll lll l ll ll ll lll llll l ll ll l ll lll ll ll ll lll l ll ll l llll lll ll l lll l l llllll ll lll lll ll lll ll l lll ll l lll lllll lll lll l ll l l l lllll l lll lll lll lll l ll llll llllll l llll lll lll l llll l l ll lll ll l ll llll ll ll lll lll l llll l lll ll l ll l ll l lll l l ll l lll llll l lll ll lll ll l l llll l lll ll l ll lll l llll l llll ll lll l lll lll l l l lll lll l ll l l llll l llll lll ll llllll l ll ll l lll llll l lllll lll l ll llll lll l ll ll ll ll ll llllll lllll lll ll ll ll lll l ll lll lll lll l llllll ll lll lll ll ll lll l ll llll lll l lll l ll lll ll ll ll llll ll ll ll llll l llll ll lll ll lll l ll lll lll llll ll l l ll l ll ll llll lll l lll l l ll l lll ll ll ll llll ll ll ll ll ll l ll lll l l lllll ll lllll ll ll ll ll l ll ll ll l ll ll l llll ll ll l lll l ll l l llll ll ll l ll llll lll l ll l llll l lll lll lll l l ll ll ll l ll llll llll l lll ll ll ll ll l ll ll lll lllll lll ll lllll ll l l lll llll lll l lll ll ll ll ll l l ll ll ll l lll l l ll ll ll ll lllll ll l lll ll lll ll ll lll l lll ll ll lll llll lll ll ll lll lllll ll l ll llll l ll llll l ll l ll lll llll lll lllll ll lllll ll ll lll llllll l ll ll lll lll lll llll l lll ll lll ll ll ll l l l llllll ll lll l lll lll l ll llll ll lll l lll ll ll lll lll lll llll ll lllll ll llll ll lll lll ll l lllll ll l lll l lllll llll ll l lll ll l ll lll l l lll l llll llll l lll ll ll l ll ll ll lllll llll ll l ll l llll l l ll l ll llll ll l lllll ll ll l lll l lll l ll lll ll ll lllll l l lllll ll ll l ll llll ll ll lll lll ll ll l llllll ll l lll lll ll ll l lll lll l ll l ll ll lll l ll l ll l ll ll ll l ll l llll llllll ll lll lll l ll llll ll lll ll ll l lll llll lll ll ll l lll ll ll llll llllll lll lllll ll l ll ll ll llll lll lll ll l ll lllll ll ll l l ll ll lll l ll ll l l l llll l lll lllll ll lll ll ll lll lll l l lll llll ll llll l lll ll l lll lll lll ll l ll lll l llll ll ll lll ll l ll l llll lll l lllll l ll ll ll ll ll ll ll l ll l ll l lll l ll llll l l ll ll ll l llll llll ll llll l l ll lllll l l lll ll ll l ll l llll ll lll l ll lll ll llll llll lll llll ll lll ll l ll lllllll ll lll lll llll lll l ll ll l ll l lll ll llll ll lll ll lll lll ll l l lll l l ll ll ll l lll llll lll lll lll ll l lll ll llll ll l l l lll lll ll l ll ll lllll ll lll lll ll ll ll l ll lll ll l ll l llll ll lllllll l lllll l ll ll l ll ll l lll l ll lllll l lll l ll l ll ll llll llll l ll ll lll ll ll l ll ll lll l ll lllll l l lll ll lll ll ll ll llllll lll ll ll lll lll l ll lll ll llll l ll lllll l l llll ll lll l l ll lll llll llll ll l ll ll llll ll lll lll l l lll ll lllll llll ll llll l llll l ll ll llll llll lllll lll lll lll l lll ll ll l ll llll l lll ll llll llll ll l ll l llll ll lll l lll ll ll ll l llll ll lll llll l llll lll ll ll llll ll l ll l ll l lll l ll l l l ll lll lll llll l lll ll lllll l ll l ll ll ll lll l l ll ll lll ll ll ll llll l ll l lll l ll llll l ll lll ll l l ll ll l lll lll ll ll ll l lll llll ll l lll lll lll l ll ll l lll l ll ll ll ll l ll lllll ll ll l ll ll llll ll ll lll l llll ll ll l ll l ll llll ll ll l ll lll ll l lll ll l lllll ll ll l ll ll l ll ll ll l ll ll llll ll ll lll lllll l ll l ll l ll l ll ll ll lllll ll lll l ll llll l ll ll lll lll ll ll llll lll lll llll ll ll ll lllllll lll l ll ll ll llll ll lll ll lll llllll ll l ll lllll lll ll l l ll l ll ll l ll ll ll ll l l lll l lll lll llll llll l ll ll l ll l lll l lll lll l llll llll l l ll ll l l ll l ll ll ll ll ll ll lll ll ll l ll ll ll ll ll llll ll ll ll ll llll l lll l lll l l ll lll l ll lll l ll lll l l llll l l ll ll l ll ll llll l ll ll ll l lll ll ll ll ll l ll ll llll ll ll l l llll ll ll ll ll lll ll lll lll lll l lll ll ll l l llll lll ll lll ll l l ll l ll l ll ll ll llll ll ll lll lll llllllll l ll ll ll ll ll l ll ll l l ll ll ll ll l ll l lll lll ll lll l lll lll ll l lllll lll l lll ll llll ll lllll ll lll l lll l ll llll lll lllll ll lll llll lll l lll llll l lllll l lll l ll lllll ll l ll lll lll lll ll l ll l llll ll ll ll ll l l ll l ll llll ll llll l ll ll ll l ll l ll ll lll ll ll l lll llll lllll l ll ll lll l l llll ll llll l llll ll ll ll lll llll ll l ll lll ll lll llll ll l l ll ll lll ll l ll ll lll l lll ll ll l ll lll l llllll lll ll ll lllll lll ll ll lll ll llll ll l ll ll ll lll llll lll lll l ll l lll ll l l llll l l llll ll l lll l l llll l ll llll lll l lll ll lll llll llll ll lll l lllll l ll ll lll l ll l ll ll llll l ll l llll ll lll ll l llll lll ll ll l ll lll l ll llll lll ll lll ll ll lll llll l llll l lllll l l lll ll ll l llll lll l l ll llll ll ll ll llll ll ll ll lll ll l ll l ll l l llll ll l l llll ll l ll l lll llll ll llll l ll ll l l ll ll llllll llll ll l ll llll llll ll ll l ll ll ll lll lll ll lll ll llll lll ll ll ll lll ll l llll ll ll l l ll lll lll l l l llll l lll l ll l ll ll llll llll ll l l lllll ll lll lll llll ll ll l lll l lll ll llll l l llll lll ll ll l ll ll l ll lllll l lll lll ll l ll lllll l lll ll ll ll lll l ll lllll l l ll l ll l ll ll l ll ll ll l ll ll ll ll llll ll l lll ll l lll l ll llll l ll lll l ll ll ll l lll l lll ll ll l ll ll lll l ll ll lll ll lll lllll lllll ll ll l lll l ll lll l lll ll ll l ll lll l llll ll ll lll lll l ll lll lll ll llll ll ll lll lll llllll l lll ll lll l lll lll ll lll llll ll ll lll lll l l llll l lll llll llllll ll l ll ll ll ll ll ll ll l llll lll l ll ll lll ll ll lll llll l ll l ll lll l ll ll l lll l l ll llll l ll ll ll lll l l lll llll ll l ll lll l ll ll llll ll ll lll lll ll l ll lllllll ll lll ll lll lll llll l ll l l lll ll ll llll llll ll ll l lll ll llllll l ll l ll lll ll ll ll llll l l lll llll l l lll lll ll l ll llll lll lll llll ll lll llll ll lll lll lll lll lll llll ll llll lll l lllll ll ll ll lll ll ll ll ll ll ll ll l l l ll lll lll lll lllll llll l ll lll lll l l ll l ll l lllll l lll l l l lll l ll llll llll ll ll l ll ll l l ll ll ll ll lll ll ll ll lll ll llll ll lll ll ll lll l llll l lll llll ll lll l lll lll l ll l l llll lllll ll l llll l ll lll l l ll lll ll llllllll ll lll ll ll ll llll l llll ll lll ll ll ll lllll l ll l llll ll l l lllll lll l l ll ll llll l ll ll l lll llllll ll ll ll lll lll ll llll ll l ll llll l lll lllll ll l llll ll l llll l ll l ll llll lll l ll lll ll lllll ll ll lll lll l ll ll l ll llll llll l ll ll l lllll ll lll llll ll ll ll llll l lllll lllll ll lllll ll ll ll l l ll lll l lll l l lllll lll l l lll l lll l lll ll ll lll l ll ll lll l l lll l ll l llll lll ll ll ll lll l l lll l ll ll l lllll ll ll ll l ll ll lll llll l l lll ll lll lll l lll ll l ll ll ll ll l ll ll l llll ll lll ll ll ll ll llll l l lll l l l ll llll llll lll ll l ll lllll lll ll ll llll llll l l ll l lll llll l ll llll l lllllll ll l llllll ll l lll lllllll ll ll ll ll ll l lll ll llll ll lll l ll lllll ll lllll l ll lllll l l l lll ll lll lll l lll lllll lll l ll ll llll l ll l lll ll lllll ll l ll l l lll lll l l lll llll lllll l lll l lll l ll ll ll l ll ll ll lll ll ll llll ll l ll l ll ll l ll ll lll l lll ll lll lllll ll ll ll lll l llll lll lllll l ll l l l lll ll l lll ll l ll lll l l lll ll llll ll ll llll ll lll l l ll ll ll ll ll l ll l lll ll ll llll l lll ll ll l ll l ll lll llllll llll ll lll l lllll ll ll l l lll lll ll l l ll ll lllll ll ll lll lll l lll lll ll ll llll ll l llll ll l ll lll ll lll ll llll l lll l ll l llll l ll ll llll llllll ll ll l l lll l lll ll l ll lll ll lll ll lll llll lll ll llllll ll ll lll ll l l lll l lll llll l lll ll l ll ll l lll llll lll ll ll lll lll ll ll llll ll lll ll l l llll l ll ll ll l lll lll l l l llll l ll lll lll l llll llll llll l ll lll l lll lll l lll llll l ll ll ll ll ll ll ll ll lll ll ll ll l lll ll ll ll ll l ll ll l lll lll ll lll l ll lll l lll l ll l ll l ll ll lllllll lll lll l lll llll ll l ll ll ll ll l lllll l ll ll l lll l ll lll l lllll l ll l lll l ll llll ll ll l ll lll ll l ll llll ll ll l lll ll llll lll llll lll l ll ll lll ll llll ll ll llll lll l lllllll llll ll ll ll lll lll ll lll ll ll lll lll lllll ll l lll llllll l lll l ll ll ll l lll llll ll llllll lll ll ll lllll lll ll lll l ll llllll ll lll llllll lllll l l lllll l ll lll l l lll ll l ll l llll ll l l ll l lll l lll lll ll l l ll ll lll l ll ll l llll l lll l ll llll l ll l ll ll l llll l lll lll l lll l ll lll ll lll ll ll l ll ll ll lll lll lll ll l lll ll l l l ll l l l ll lll l llll ll l lll lll ll ll lll lll lll ll lll l l lll ll ll llll l ll l l l lll llll ll ll llllll llll lll ll l ll ll llll ll ll l ll lll l ll l ll l ll lll lll ll lll ll l lll ll l lll ll l ll ll ll ll ll ll ll ll ll ll ll llll lll lll l ll lll ll llll llll l lll llll l lllll ll l ll ll ll ll lll lll l ll ll ll l l lll l l lll l l ll l lll l ll ll llll lll l ll ll l l ll l ll llll l ll lllll ll l ll ll llll ll ll llll lll l l l l ll llllll lll ll ll ll l ll lll l lll ll l lll l lll l lllll lll ll lll lll lll ll l lll l ll lll l ll lll ll llll l ll ll l ll l llll l lll lll lll ll lllll ll l lllll lll ll lllll lll ll ll ll l llll l ll l lll ll l lll l llll ll ll ll ll lll lll llll l lll lll l l lllll lll llll l l llll lll ll l lll l lll l l lll lll ll lll lll lll lll l l lll l ll lll ll ll l llll ll l lllllll ll ll llll llll l lll ll l l llll ll l ll lll llll llllll llll ll l ll lll ll ll l l l ll ll llll lll lll llll ll ll llll l ll l ll l ll ll l ll llll lll lll l ll lll lll lll l lll ll lll ll ll ll lll lll ll lll lll lll llll −4 −2 0 2 4 − Quantiles of standard normal Q uan t il e s o f l og ( U ganda ) ll llll ll ll lll l ll lll l lll ll ll lll ll ll l lll lll lll lll ll lllll llll l ll l ll ll ll ll l llll l l ll ll lll ll l ll l ll lll l lll lll llll lll l ll llll lll ll lll lll l l llll ll ll lll ll l llll ll llll l l ll ll l l l ll ll llll llll l ll ll ll l ll ll l ll ll ll l l lllll l ll l lll l llll ll lll l ll lll l lll lll llll ll ll l ll ll l ll ll l lll l lllll lll l ll lll l l llll lll ll lll lll lll lll l ll ll ll ll l lll l l ll l ll l ll ll lll ll ll l lllll ll lll llll lll l lll llll l llll l lll ll ll ll lll l l ll lll l lllll lll l llll llll l l ll ll lll l lll l ll llll ll ll llllll lll l lll l lll lll ll l lll ll l ll ll llll l lll lll ll ll ll l llllll ll l lll ll lllll lll lll ll llll ll ll l l ll l l lll ll l llllll l ll ll l l ll ll llll lll ll l ll ll l lll llll ll llll ll l lll l ll l lll l lll ll l lll lll ll lll llll ll ll ll l l ll ll llllll lll ll llll lll ll ll l lll llll ll ll ll ll ll l lll lll l l lllll l ll llll llllll ll lll lll l lll l ll lll l lll ll l ll ll llll ll l lll lll lllll lll ll l ll lll l llll l ll lll ll lll l ll ll ll ll lllll lll lll ll l llll l lll llll l ll llll lll ll ll ll ll l llll l l ll ll l ll lll ll ll lll ll l ll ll ll ll lll ll l lll ll ll llll l l lll ll lll l lll ll lll l ll lll lll ll l ll ll l ll llll lll ll lll ll lll llll l ll ll llllll lll l ll ll l lll ll l ll l ll lll ll lll ll ll lll ll llll lll lll ll lllll ll lll lll ll lll lll ll lll lllll ll l l ll ll lll ll l ll lll l ll l lll lll l l l llll l ll l lll llllll ll ll l lll lll l ll llll ll llll ll lll ll ll lll lllll llll lll llllll l ll llll lll ll lll l ll ll ll lllll l l ll l ll lllllll ll ll lllll lll lll lll ll ll l ll lll l ll ll ll ll ll ll l ll lll ll ll ll l l l lll lll lll llll lllll llll ll lllll llll l lll l ll lll ll ll l ll lll l llll llll l lll ll ll ll l ll ll lll ll lll ll l ll ll l l ll ll ll ll llll ll ll l ll ll llll l ll lllll ll lll lllll ll l llll ll l l llll ll ll ll lll l l lllll l llll l ll ll ll l ll l l lll ll ll ll lll ll lll lll lll lll ll lll ll ll ll ll ll ll l l ll ll ll ll ll ll lll lll lll llll ll ll ll ll l ll llll ll l l ll ll ll l llll ll lll ll lll ll ll lll lll ll lllll lll lll ll l llll l ll lll ll lll l l lll ll lll ll l ll ll l l ll l lll llll ll lll l ll l llll lll lll l llll ll ll ll llll lll ll lll lllll l ll llll l lll ll ll l llll ll l ll l ll lll lllll ll l llll ll lllllll lllll llllll lll llll l l ll ll lll l lll ll ll ll l lllllll l ll ll ll ll ll ll l ll ll lll l l lll l lll l lllll lllll ll l l lll l ll llll l ll ll l l lll ll l ll ll lll ll ll lll l lll ll ll ll l ll l ll llll ll l l lll lll ll lll lll ll lll l ll ll ll lll l lll l ll ll lll lll lll ll llll l l llll l ll ll l ll l lll l ll ll l ll l l lll l l lll ll l llll lll llll llll l ll lllll ll ll ll l lll llll l llllll l ll l ll l ll l ll l llll lll ll ll l ll ll ll l ll lllll l 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lll l lll l llll l lll ll l lllll ll lll lll ll ll ll lll l lll lll l ll ll lll l ll lllll l ll lll llll ll ll ll lll ll l llll lll ll l ll lll ll lll ll llll ll l ll l lllll ll ll l ll ll ll llll l lll lll ll ll lll lll l ll llll l lll ll l llll ll ll lll ll ll ll l l lll ll l lll l llll llll ll lll ll l l ll ll llll l ll l llll l ll ll lll lllll l l ll l lll llll llll l ll ll l lll ll ll l ll lll l l lll ll lll ll l lll ll lll lll l lll ll l lll ll ll l ll lll llll ll ll ll llll l ll lllll ll llll ll ll ll ll lll l lll ll lll l lll llll ll lll ll l l lll llll ll ll l ll ll ll l llll l ll ll l llllll ll ll ll lllllll l ll lll ll l ll lllll ll l lll ll lll l lll l l llllll l llll llll ll lll ll ll lll lll lll l llll l ll ll l lll lll lllll ll l ll ll ll l ll llll llll l lllll lll l lll llll lll ll ll ll lll l ll l ll l l ll ll l lll ll l llll ll ll ll llll l llll ll ll ll l ll lll ll ll lll l ll ll llll ll l llll l lll llll lll llll ll l ll l lll ll ll lll ll ll llll lll ll l ll ll lll 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ll lll ll l lll llll lllll ll lll l llll ll ll ll ll ll ll l llllll lllll llll lll lllll ll lll l ll l ll lll l ll ll ll ll ll ll lll lll ll l lll lll ll ll ll lll l l lll l llll l ll lll ll l lll l ll ll l ll lll l ll l ll lll l ll l l lll lll l ll l lll l llll ll l ll ll lll ll lll llll l lll lll l llll l lll ll l llllll lll ll llll ll l lllllll ll ll ll l ll ll l l llll lll lllll ll l lll ll l lll l lllll l ll lll l ll lll ll ll ll ll l ll lll l ll l lll ll ll ll ll l ll lll llll lll lll l ll l ll l lllll ll ll lll l lll lll l ll llllll llll ll lll lll ll ll ll l lll l ll ll l lll llll ll l lll ll l ll l lll lll llll ll llll ll l l lll lll lll lll l ll ll lll lll lll llll ll lll ll l ll l ll ll llll ll ll llll ll l l llll l lll l l llll ll lll l ll ll l ll ll llllll lll ll ll lll lll l l lll ll ll ll ll llll lll lll lll lllllll l lll ll ll l ll l ll lll ll ll l ll lll lllll ll l llll l ll l lll ll lll ll lllll lll ll l ll lll ll l ll lll llll lll ll lll ll l llll llll l lll ll ll llll l ll llll l l ll l l lll ll l llll l llll ll lll l ll lllll ll l ll lll l lllll ll ll ll l ll l lllll lll lll l lllll l ll ll lll ll ll l ll llll ll lll llll llll ll ll ll l l l ll l lll l llll ll l llll ll ll ll l llll ll ll ll l ll ll l lllll ll llll lll llll l ll ll l lll ll llll llll llll l lll lll ll ll l ll ll l ll lll l lllll l ll l ll ll lll lllllll ll lll ll l ll ll llll ll ll ll lll ll l ll ll llll ll l ll l ll llllll llll ll ll llll l llllllll l ll l l ll lll ll ll ll ll ll llll ll l lll l llll l l l l l ll ll lll ll l ll ll ll llll llll ll lll ll ll lllllll ll lll llll ll l lll l ll ll lllll l llll lll lll llll l ll llll l ll ll ll lll ll l lll l ll l llllll ll lll lll ll ll l lll ll lll l lll ll ll llll l l lll ll lllll l ll l lll lll ll ll l lll ll ll l llll l llll lll l lll ll ll ll lll ll l ll l lll lll l ll l lll llllll l ll ll lllllll llll lll lll l lll ll ll ll ll ll l ll ll l lllll l ll l ll ll lll l ll ll ll ll ll l ll ll lllll llll l l ll lll l lll lll l ll ll lll ll ll ll lllll ll ll lll ll l lll ll lllll ll l ll llll l ll l ll lll ll ll ll ll l ll l llll l llll ll ll lllll ll llll ll lll ll ll ll l ll llll lll ll lll ll ll l l l ll lll ll lll l lll ll l ll lll ll llll lll ll ll ll l lll ll ll lll ll llll lll l ll l lll ll ll l lll l llll llllll ll ll l lll lll l ll l lll lll llll ll ll llll ll l lll lllllll l lll lllll ll lll ll lll ll l l ll l ll ll ll ll lll lll ll ll ll lll ll ll lll l ll ll l llll l lll lll lll llll l l l l lll ll llll ll lll ll ll l llll l ll l l ll ll ll ll ll l lll ll lll ll ll llllll lll l ll l lll lll l lll lll l ll l lllll ll lll lll ll ll ll ll ll lllll lll l l llll l ll ll llll ll l llll ll ll l l l ll l lll llll ll lllll l llll lll lll llll lll ll l llll ll lll llll ll lllll ll l ll lll ll lllll l lll l l −4 −2 0 2 4 − Quantiles of standard normal Q uan t il e s o f l og ( U ganda ) l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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l l l l −2 0 2 4 6 − Quantiles of log(Uganda 2002) Q uan t il e s o f l og ( U ganda ) Figure 2: QQ-plots of standard normal quantiles against standardised log-transformedUganda expenditure data for 2002 (left) and 2005 (middle), as well as QQ-plot of stan-dardised log-transformed Uganda expenditure data for 2002 against 2005 (right).The estimated growth incidence curve shown in Figure 3 is close to 0 on the whole19nterval (0 , . − . . . Probability G I C − . . . Probability G I C Figure 3: Growth incidence curve for the Uganda data from 2002 to 2005 with 95% con-fidence bands and national poverty line. Simultaneous confidence bands are shown in theleft plot, while pointwise confidence bands with the World Bank algorithm in the rightplot.Let us now consider the expenditure data from 1992 and 2002. Inspecting QQ-plots ofstandardised log-transformed data shown in Figure 4 we find that both data sets are notlog-normal and distributions of both data sets differ from each other not only in location20nd scale. Hence, for the plug-in confidence bands correction c s needs to be estimated. l llllll lll ll lll lll lll lll ll l l llll l llll lll l ll l lll lll ll l llll l ll lll l ll ll ll ll ll ll l l l lll ll llll ll ll l ll ll lll l ll lll l lll lll l l llll l lll lll llll ll ll ll l ll ll ll l lll l lll l ll l ll l llll ll ll lllll llllll lllll lll ll l ll lll lll lll ll lll ll llll l lll l lll lll ll ll l ll lll l l llll ll lllll lll lllllll llllll ll llll l ll l l ll ll l lll ll lll llllllllll l ll llll lll llll l l lll ll l l lllll lll l ll l ll llll l lll ll llll l l llll ll ll l llllll ll ll ll ll l l llll llll l llll ll lll l ll llllll ll l llll l ll l ll ll lll ll l ll llll ll ll lll ll l ll lll ll ll lllll lll ll l ll l llllll l lll l llll ll l ll lllll l ll lll lll ll ll ll ll lll lll lllll lll l lll lll lllllll l ll ll l ll ll ll ll l ll l lll l lll ll lll lll l l lll l l ll l ll lll llll lllll l ll lll ll ll l ll lll ll l ll llll ll ll ll l ll llll l ll lll llll lll ll ll ll ll llll lllll l llll l l llll l ll lll lll l ll lll l lll ll l ll ll lllll lll lll llll ll llll l lll llll lll ll l ll ll lll lll ll l llll l ll l ll lll llllllll l ll ll l ll llll ll l lll lll llll l l ll llll ll ll ll l ll lll ll l ll ll ll l llll l lll l ll lllll ll l l l ll lll ll ll ll lll l l l ll lll ll llll lll ll l ll ll ll l ll lll lll ll ll l ll l lll lll l ll l ll lllll l ll ll ll lllll ll l lllll ll ll ll l ll lll ll ll lll lll llll ll l lll lll ll ll lll ll lll l llll lll lll lll ll l lllll lll l l ll lll l ll l ll llll l ll l lll lll l llll lll ll l ll ll l ll ll llll lll lll llll ll l ll l lll ll ll lll ll l ll lll lll l lll l lllll lll l ll lll ll l ll lll llll l lll l l ll llll l lll l ll l ll ll l ll ll ll ll ll l lll ll ll lll l lll ll l llll ll l lll lll l l lllll ll lll lll llllll lll lll ll lll ll ll l ll l l ll ll ll lll l lllll lllll llll l l ll ll l lll l ll llll l llll l lll ll ll l lll llll ll lll lll ll llll ll ll l ll ll lll ll lll l lll l ll l l ll l l lllll ll llll l ll ll lll llll lllll ll l l ll ll ll ll l llllll l ll l ll ll ll llll ll l ll ll l lll lll l lll lll ll ll ll lll ll ll lll lll ll ll ll llll ll ll ll lll lll l ll lll ll l l llll l lll llll ll l lll lll ll ll ll ll lll ll l llll ll ll ll ll ll llll lll l lll llll llllll lll l lll ll llll l lll ll ll llll ll ll l ll lll l l l l lll ll lllll lll llll ll l lll ll lll l l ll ll llllll ll l lll ll ll lllll ll l llll lll ll l lll ll ll l ll l llll lll l lll lll llll lll lll l l ll lll lll l ll lll ll ll ll llll l ll l lll lll l lll l lll llll lll lllll ll lll llll ll llllll llll ll l lllll ll l llll ll ll ll lll ll l lll llll l lllll ll lllll ll l l lll llllllll lll l lll l ll l l ll ll llll ll ll ll ll l l ll lll llll lllll l lll lll lll llll ll ll lll l ll ll llll ll ll lll ll lll l ll l llll l llllll l llll l lll lll ll l ll ll lll ll llll lll llll ll ll lll lll lll ll ll l llll ll ll ll lll lll l lll ll l ll l ll lll llll ll l lll ll llll ll ll llll llll lll ll l lll l lll lllll lllll l l ll lll llll lll l ll lll lllll l ll l ll lll l ll ll llll llllll ll l ll ll ll l lll llll llll llll ll llll lll l lll llll l l lllll ll lll ll l lllllll ll ll lll ll l llll l ll l ll lll ll l lll ll lll ll lllll ll lll l l l llll l llll ll lll ll llll ll llll llll lll l ll l lll l llll ll lll ll ll ll l ll l ll l l lll ll l ll lll lll ll l llll llll ll lll l llll lll l lll l ll lll ll ll ll l lll ll ll lll llll llll l lll ll ll lll l lll l ll ll l lll ll lllll l lllllll ll ll l ll l lll l lll lll l ll lll l lllll lllll l ll lll l lll ll ll lllll lll ll llll ll ll ll ll lll l lll l ll ll lll lll ll lll ll ll lll lllll lllll ll llll l l llllll ll ll ll lll llll lll ll lllll l ll ll ll ll ll lll lll ll ll l lll l ll ll lll ll l lll ll lll l ll l l lllll llll l l lll ll l llll ll l ll lll l ll ll l ll ll ll ll l lll ll lll lll ll ll l ll l llll l l ll lll llll ll llll lll l llll l ll ll ll ll l ll llll ll lll lll ll llll llll llll ll ll l ll l llll lll ll lll l llll ll ll ll ll ll ll llll ll l l lll ll l llll l llll l llll l l l lllll l ll lll l ll ll lll l lll llll l ll llll ll lll lll ll ll ll ll ll ll ll ll lllll llll l ll ll ll ll ll ll lllll l l lll ll ll ll l ll lll llll llll l lll ll lllll ll lll l llll ll ll l ll lll llll l lllll ll l lll l llll llll llll ll lll l l lll ll ll ll ll ll ll ll lll l lll l ll lllll l lll lll l lll ll ll lll l l lll l ll llll llll lll l llll l lll llll lll llllll l lllll l l lllll llll ll ll llll ll l ll lllll ll ll lll ll llll ll llll l lll l lllllll ll lll l lll llll ll ll ll ll l l lll lll lll l l llll lll lll ll ll ll ll l lll lll l ll ll ll ll ll ll ll lll ll ll l lll ll ll l l llll l lll ll l ll ll ll lll l ll ll llll ll l ll ll ll ll l l l ll lll lll ll lll l ll ll ll ll l ll lll l l lll lllll l lll l ll llll lll ll ll lll ll l llllll lll llll ll lll l ll ll lll ll lll l lllll l ll llllll ll l ll lll llll ll l lllll ll llll lll ll l ll ll lll lllllll l l ll ll l l ll lllll lll ll l ll ll llll ll l llll lll llll lll l ll ll ll lll llll llllllll ll l llll ll l ll ll l l l llllll lllll ll ll ll l ll llll ll ll l l llll ll ll ll ll lll l llll l llll ll ll lll lll lll llll lll l ll l lll l lll lll l llll lll ll lll ll ll ll lll llll lll l lll ll lll lll l ll ll llll lll ll llll lllll llllll lll lll l ll l llll l l lll l ll lll lll llll ll llll lll l ll ll l ll ll l ll llll ll lll lllll ll l lll ll lll l llll l llll l ll lll lll lll l ll lll l lllll ll ll lll lll llll l lll ll ll ll ll l l ll l l ll lll l ll lll ll llll lllll lll l ll l l ll l l ll ll ll ll llll l lll ll lll ll l l ll l lll lllll ll ll ll ll ll ll l ll llll ll l ll llllll ll llll l lll lll ll ll llll ll llllll llll ll l lll ll lll lll ll ll ll lll ll ll llll lll lll l ll llll lll ll l ll l ll l llll l ll lll ll ll lll l lll ll l lll l ll lll ll lll ll llll llll ll ll llll l l ll l llll lll l ll ll ll lll ll ll ll ll ll l ll ll ll lll llll ll ll l ll lll lll l l ll ll ll l lll ll lll ll l l lll ll lll ll l ll llll l lll ll llll l ll lllll ll ll ll ll lll l ll lll ll l lll ll lllllll ll lll l l ll lll ll l lllll llllll l l ll ll ll lll lll ll l lllll ll lll ll llll llllll l ll l lll ll ll ll ll llll ll llll l lll l lll ll ll llll l l lll ll l ll llll ll ll ll l lll l llll llll llll ll l ll lll ll ll llllll ll l ll lll lll l ll lll l lll ll ll l ll ll ll lll ll ll ll ll l lll l lllll ll lllll ll l lll l lllll ll l lll ll l ll llllll ll lll ll lll l ll ll ll ll l lll ll ll ll l ll lll lll lll ll ll ll l l l lll l l ll lll l l ll ll ll ll lll lll lllll lll llll ll lll ll lll lll ll lll ll l ll ll l ll l lllll llll ll lll ll l lll llll l ll ll ll l lll lll lll llll ll ll ll llll l l ll ll lll llllll ll ll ll l ll lll ll l ll l ll l l l lllll l l lll lllll l llll l l lll l ll ll lll lll l ll ll llll lll l llll lll l ll lllll ll l lllll l ll ll l ll lllll l ll lll l l ll ll lll ll lll lllll ll ll lll l ll lll ll ll lll llllll ll llll ll lll ll llll ll ll lll ll lll l ll lll l lll ll l lllll ll l ll ll lllll ll lll 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ll ll ll llll lll ll ll lll l lll ll lll lll llll l ll l ll ll ll ll llll lll lll lll l ll lll ll lll lllll l lll llll lll llll ll ll l lll l lll ll l ll ll l ll ll ll lll lll lllll ll l ll ll l llll ll l ll ll ll l ll l ll ll ll l lll lll ll l ll ll ll lll ll ll llll lll lll ll l l llllll lll ll lll lllll ll l lll ll llllllll l ll lll l ll l l ll llll lll lll lll lll ll ll lll llll ll lll lll l ll l ll ll ll llll l ll ll ll lll lll ll ll ll ll ll l ll lll ll ll ll lll ll lllll llll lll llll ll ll ll lll ll l ll l ll ll l ll l l llll l ll lll llll lll ll lll l lll l l ll llll l lll ll l ll ll l lll llll l lll ll ll l ll ll l ll ll ll llll l ll ll ll l lll l lll l ll ll l lll l ll lll l l ll ll ll l ll lll ll l lll llll ll ll ll ll ll lllllll lll l lll l ll lllll lll llll l lll ll llll ll llll ll ll l ll ll l ll ll ll ll ll l ll llll ll ll ll lll lll ll l lll l lllll l lll lllll llll ll ll llll l lll llll ll ll l lllll ll ll l lll l ll lllll ll l l l lll l ll ll llll ll ll l l ll l ll llll lll l l lll ll lllll l lll l l l ll ll ll lll ll ll ll lll llll lllll ll ll l ll lll ll lllll ll ll llll l l lllllll l lll ll l ll ll lll ll ll l ll l llll l llll llll ll lll l lllll ll l ll l lllll ll ll ll llll ll l lll ll llll l llll ll l ll ll ll l llllll ll l lll l ll lllll ll ll lll ll l l lll ll ll ll l ll ll l l lll l llll lll llll ll lll ll llll lll lllll ll llll lll l lll lll ll ll ll l lll lll ll l ll lll ll ll l ll ll ll ll lll l l ll ll ll lll l lll l ll l ll lllll ll ll ll l ll l ll l ll ll l ll l ll lll lll lll l l l ll ll ll lll l ll ll ll l ll l ll ll ll lll ll ll ll llll ll lll l lllllll ll ll ll l ll ll l lllll l ll ll ll lll lll llll lll ll ll ll l ll llll l ll lll l lll ll l llll l ll ll ll l ll llll l ll ll l lll ll ll llll lll l l ll lll lll ll lll llll ll llll l ll llll llll ll ll ll ll ll lll l lll ll l ll l ll l lll ll lll ll ll lll l llll ll ll ll lll ll llll lll llll llll ll l l llllll llll l ll llll l ll ll l ll ll l ll llll lll ll ll ll lllll lll llll lll ll llll l l ll lllll ll ll l l lll l ll ll lll l ll lllll ll lll lll lll lll l lll l lll ll lllll ll l ll llll ll lllll ll ll lll ll ll ll l ll ll lll ll lll lll lll lll ll lll llll lll llll ll ll l lll ll lll lllll l ll ll l ll lll l llllll lllll l ll l llll lll lll lll l l ll l ll lll ll lll l lll ll ll ll ll ll l l ll ll ll l ll lll lll ll l lll ll l ll ll ll ll lll l ll ll l ll lll l ll ll l l ll l l llll ll lll l llll l ll lll l ll lll l ll l ll ll ll ll l ll l llll l l ll lll ll l l lllll lll lll ll lll l l llll llll l ll lll ll l l ll l l ll ll lllll lllll ll ll ll l lll llll ll ll ll lllll ll llll ll l ll lll l llll l lll llllll lll llll ll lll ll l lll l lll ll lll llll llll lll ll ll l llll ll l ll ll ll ll lll l ll lllll l ll ll ll ll ll l ll lll ll ll l lllll lll lll llllll l ll lll l l l ll llll llll ll l l ll l lll lll l l ll ll l lll llll ll ll ll ll l ll llll ll lll lll lll ll l l l llllll l lllll ll l l lllll l l llll l ll l llll lll ll ll lll lllll l l ll ll lll ll l ll ll ll l ll l ll ll ll ll lllll ll lll ll l llll ll lll ll lll lll lll ll ll lll ll ll ll ll ll lll llll l l llll lll ll l lll ll l llll llll lllll ll ll l ll ll ll l lll l lll l l llll l lll ll ll l lll l ll ll l ll l ll ll llll lllll ll l lllllll l lllllll l lll lll l ll ll ll llll ll ll ll ll l ll lllll llll l lll lll l l lll ll ll ll l ll ll llll l ll lll l ll ll lll l ll ll ll l lll ll l ll lll l ll ll ll ll l ll lll lll ll l lll lll ll lll ll ll lll lll l lll ll ll l llll lll ll ll lll lll lll l ll l lll ll l llll ll lll ll ll ll ll ll ll lll l lll ll lll ll lll ll ll lllll l ll l l ll ll l l llll lll ll ll lll l lll l lll ll l l ll lll ll lll llllll ll ll l lll l ll lll lll ll l llll l ll lll lll lll ll l llll l l lll ll ll l ll l lll l lllll llll ll l l lll llll lllll l lll ll ll lllll ll l ll l l lllll ll l ll lll ll ll ll lll ll ll ll l lll llll lll lll l lll ll ll ll ll ll lll lll lll llll ll lll lll ll lll llll l lllll ll lll llll l lll l lll lll lll ll lll ll lll ll l lll l lll lll ll lll ll l ll ll ll l l ll l llll l ll l ll ll ll ll l llll l ll lll ll ll lll ll llll l ll llllll ll ll ll llll l l ll l l lll l ll lll l l ll lll ll lll ll lll ll ll ll ll l llll l l lll ll ll l llll l llll ll ll llllll lll ll ll ll l ll ll llll l llll ll ll lll ll lll lll lll ll llll l lll ll lllll llll llll ll lll ll ll ll l lllllll lllllll ll lll l llll lll l lll ll lll l llll l ll ll ll ll llll ll lll l ll ll ll lll lll ll ll ll l llll llllll ll lll lll l lll llll ll llllllll ll ll llll ll l lllll l ll ll lll ll ll ll ll l llll ll lll lll ll lll ll l lll llll llll l ll l ll l ll lll l ll llllll lll ll l lll ll ll ll llll ll ll l lll ll lll ll l lll llll ll lll l l ll ll llll llll ll ll ll llll lll ll lll ll llll ll llll l lll ll l l ll lll lll l lll ll ll ll lllll l lll l l l ll ll ll lll llll l l ll ll ll l l lll ll l lll ll ll lllll l lll lll ll l ll lll ll l ll l ll lllllll lll ll lllllll ll lll lll lll l l llll lll lll llll ll l l ll lll lll ll ll lll l ll ll l l lll llll llll lll l lll l lll llll l ll l lll lllll llll ll lllll ll l lll l ll ll lll l l ll l ll ll lll ll ll llll lll ll ll lll l ll ll ll lll lll ll lll l lll l l llll l lll llll ll ll ll l lll l ll l ll llll lll llll lll lll ll ll ll l lll ll lllll l lll ll lll ll l lll l ll l lll ll lll lll l ll llll l llll ll l lll lll ll ll lll llllllll ll lll lllll llll lll lllll l ll lll ll l lll ll l ll lll ll lll ll ll llllllllll l ll ll lll ll ll ll llllll lll ll ll ll l ll ll l lll lll l l l lll l l ll ll lll l ll ll ll l llll ll lll lll ll lll ll l ll ll l ll ll ll lll ll ll ll ll llll ll l lll lll ll lll ll ll lll l l l lll ll lll l ll ll l ll lll ll l ll l llll lllllll l l llll llll ll lll l l ll lllll lllllll lll llll ll l ll l ll ll lll ll l ll l ll l lll l l ll l lll l lll l l lllll lll lll l ll ll lll ll l ll ll llll llll lll ll lll llll l lll lll lll l l lll lll llll llllll l llll ll l ll l ll llll l ll lll l lll ll l ll ll l ll llll ll ll l ll l ll ll ll llll l ll ll ll ll ll lll lll lll llll ll l l lll lllll ll lll lllll llll ll ll l ll ll lll lll l lll lll ll l l ll ll ll l llll lll l l ll l l lll l lll lllll l l llll ll ll l ll lllll l lll ll ll ll llllll l ll ll ll lll ll ll l ll l l lll l ll ll l lll lll l lll ll l lll ll lll lll ll ll l lll ll l lll ll l ll −4 −2 0 2 4 − − Quantiles of standard normal Q uan t il e s o f l og ( U ganda ) l ll l ll ll l llll lll l l llll l ll lll lll lll ll l lll l lllllll ll lll ll ll lll ll lll ll ll l ll llll l ll l ll l ll lll ll llllll ll lllll l ll ll lll l llll ll l l llll ll ll lll lll lll l l ll lll lll l l lll l lllll l lll ll lll l ll lll ll llll l l llll ll lllll ll l llll llll lll lll ll llll lll l lll l ll ll llll llll ll ll l lllll ll ll ll l ll ll l llll ll llll l ll lll llll ll l lll l l l lllll ll lll ll ll ll l ll lll lll l lll ll ll lll l ll ll ll lllll ll lll ll lll l ll ll l lllllll l lll l lll lll lllll ll l lll l ll lll l lllll llllll ll ll l ll lll ll ll ll l l ll ll ll ll l ll lll lll l ll llll ll lll l ll ll ll lll l lllll lllll lll ll ll ll ll l ll ll ll lllll lll ll lll l ll l ll l ll ll ll ll ll l ll l ll l l lllll llll lll ll l lll lll ll llll l ll llllll lll l ll ll lllll ll ll l l llll ll l ll l l ll l ll ll ll ll llll lll l l ll ll lll ll ll ll l ll lll l ll lll lll lll lll l ll ll l ll lll ll l ll l ll llll lllll l ll ll l lllllll lll l ll l lll lll l llll ll l llll ll ll ll lll ll ll lll l lll llll llll ll ll lll lll ll l lllll ll lll l ll ll l llll lll ll l llll l lllll l llll ll l lll lll lll ll l lllll lll lll lll l llll l ll ll lll ll lll ll ll ll l ll l ll lll ll l lll l llll ll l ll ll ll l ll l l lll ll lll ll ll ll ll lllll l ll lll llll ll l lllll ll l ll ll l lll l ll ll llll lll lll ll l ll llllll ll l lll llll l ll l llll l ll l ll ll lll l lll l l ll ll ll lllll l l l ll llll llll l ll ll ll ll lll l lll l lll l l ll ll lll lll ll ll llll l ll lll l lll llll ll ll llll ll lll lll llll l lllll l lll ll l ll l ll llll lll l l ll ll ll lll l llll lll lll 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ll l lll l lll l l lll lll ll lll lll lll lll l l lll l ll lll ll ll l llll ll l lllllll ll ll llll llll l lll ll l l llll ll l ll lll llll llllll llll ll l ll lll ll ll l l l ll ll llll lll lll llll ll ll llll l ll l ll l ll ll l ll llll lll lll l ll lll lll lll l lll ll lll ll ll ll lll lll ll lll lll lll llll −4 −2 0 2 4 − Quantiles of standard normal Q uan t il e s o f l og ( U ganda ) l l l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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ll −4 −2 0 2 4 − Quantiles of log(Uganda 1992) Q uan t il e s o f l og ( U ganda ) Figure 4: QQ-plots of standard normal quantiles against standardised log-transformedUganda expenditure data for 1992 (left) and 2002 (middle), as well as QQ-plot of stan-dardised log-transformed Uganda expenditure data for 1992 against 2002 (right).Figure 5 shows annualized growth incidence curves for Uganda form 1992 to 2002 to-gether with the simultaneous confidence band (left) and with the World Bank Toolkitconfidence band (right). The estimated growth incidence curve is positive for all quantilesand simultaneous confidence band does not include the zero line. Absolute poverty wasreduced between these two periods, and growth was pro-por using the weak absolute def-inition. In addition, the growth incidence curve seems to have no significant slope for thepoor and a slightly positive slope for the population above the poverty line. This suggeststhat inequality among the non-poor increased. The confidence band gives evidence thatthe overall slope of the growth incidence curve on the interval [0 . ,
1) was non-negative.Confidence bands of the World Bank Toolkit do not allow for such inference about theslope by definition.
Motivated by the concept of growth incidence curves introduced in poverty research weconsidered the ratio of quantile functions as a tool to compare two distributions. We have21 .0 0.2 0.4 0.6 0.8 1.0 . . . . . Probability G I C . . . . . Probability G I C Figure 5: Growth incidence curve for the Uganda data from 1992 to 2002 with 95% con-fidence bands and national poverty line. Simultaneous confidence bands are shown in theleft plot, while pointwise confidence bands with the World Bank algorithm in the rightplot.developed an analytical method for calculating simultaneous confidence bands for ratiosof quantile functions and growth incidence curves. Our method requires no re-samplingtechniques and rather relies on the asymptotic distribution of the difference of two quantilefunctions and therefore readily provides simultaneous confidence bands also for the quantiletreatment effect, considered as a curve. In the application to the expenditure data fromUganda we demonstrated how simultaneous confidence bands can be used for inferenceabout growth incidence curves and showed that these simultaneous confidence bands aremore appropriate than those provided by the World Bank Toolkit.
Acknowledgments
The authors acknowledge support by the Ministry of Education and Cultural Affairs ofLower Saxony in the project Reducing Poverty Risk. We also thank Gordon Sch¨ucker for22ranslating the World Bank algorithm for GIC confidence intervals from Stata to R.
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A Appendix
A.1 Proofs of Section 2
To prove Theorem 1 and Corollary 1 we use the following standard result.
Theorem 5 (Cram´er, 1946, p. 368–369) . Let X be a random variable with cumulativedistribution function F , which is continuously differentiable at some x with F ( x ) = p and F (cid:48) ( x ) > . Let also Q ( p ) = F − ( p ) denote the quantile function, q ( p ) = Q (cid:48) ( p ) =1 /F (cid:48) { Q ( p ) } the quantile density and (cid:98) Q ( p ) the sample quantile function.(i) The distribution of (cid:98) Q ( p ) is asymptotically normal with mean Q ( p ) and variance n − p (1 − p ) { q ( p ) } for n → ∞ and for every p ∈ (0 , .(ii) If in addition F is continuously differentiable at some ˜ x with F (˜ x ) = ˜ p and F (cid:48) (˜ x ) > for p ≤ ˜ p , then the joint distribution of { (cid:98) Q ( p ) , (cid:98) Q (˜ p ) } is asymptotically bivariatenormal with expectation { Q ( p ) , Q (˜ p ) } and Cov { Q ( p ) , Q (˜ p ) } = n − p (1 − ˜ p ) q ( p ) q (˜ p ) for n → ∞ and for every p ∈ (0 , . Theorem 1 shows that the distribution of { (cid:98) g ( p ) } m can be approximated by a log-normaldistribution. 26 roof of Theorem 1 From (3) and Theorem 5, estimator log { (cid:98) G ( p ) + 1 } = m log { g ( p ) } = m { (cid:98) Q ( p ) − (cid:98) Q ( p ) } is the sum of two asymptotically normal estimators. Since X and X are independent,their sum is asymptotically normal with the mean µ ( p ) = m {Q ( p ) − Q ( p ) } = m [log { Q ( p ) } − log { Q ( p ) } ] = m log { g ( p ) } and variance σ ( p ) = m p (1 − p ) (cid:20) { q ( p ) } n + { q ( p ) } n (cid:21) . Hence, { (cid:98) g ( p ) } m is log-normally distributed with parameters µ ( p ) and σ ( p ). This provespart ( i ) of the theorem. Part ( ii ) follows in the same way from Theorem 5 ( ii ). (cid:3) Proof of Corollary 1
From Theorem 1 we have that log { (cid:98) G ( p ) + 1 } is asymptotically normal with parameters µ ( p ) and σ ( p ). Let Y = (cid:98) G ( p ) + 1 − exp { µ ( p ) } exp { µ ( p ) } σ ( p ) . Then, the distribution function of Y is given by F ( Y ≤ y ) = F (cid:104) (cid:98) G ( p ) + 1 ≤ y exp { µ ( p ) } σ ( p ) + exp { µ ( p ) } (cid:105) = F (cid:32) log { (cid:98) G ( p ) + 1 } − µ ( p ) σ ( p ) ≤ log [exp { µ ( p ) }{ yσ ( p ) + 1 } ] − µ ( p ) σ ( p ) (cid:33) = Φ (cid:20) log { yσ ( p ) + 1 } σ ( p ) (cid:21) + ø(1) = Φ (cid:20) y − y σ ( p )2 + ø { σ ( p ) } (cid:21) + ø(1) , where Φ is the cumulative distribution function of a standard normal distribution. Since σ ( p ) → { n , n } → ∞ , the results follows. (cid:3) Theorem 6 (Theorem 3.2.4 in Cs¨org˝o, 1983) . Let X be a random variable with the cu-mulative distribution function F ( x ) , quantile function Q ( p ) and quantile density function Q (cid:48) ( p ) = 1 /F (cid:48) { Q ( p ) } , p ∈ (0 , . Let X , . . . , X n be i.i.d. sample of X and (cid:98) Q ( p ) be theempirical quantile function as given in (1). Under Assumption 2 with X = X = X thereexists a Brownian bridge { B n ( p ); 0 ≤ p ≤ } such that sup p ∈ [ δ n , − δ n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q ( p ) − Q ( p ) Q (cid:48) ( p ) / √ n − B n ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) with δ n = 25 n − log log( n ) . If in addition Assumption 3 (i) holds, there exists a Brownianbridge { B n ( p ); 0 ≤ p ≤ } such that sup p ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q ( p ) − Q ( p ) Q (cid:48) ( p ) / √ n − B n ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) . If Assumptions 2 and 3 (ii) hold, there exists a Brownian bridge { B n ( y ); 0 ≤ y ≤ } suchthat sup p ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q ( p ) − Q ( p ) Q (cid:48) ( p ) / √ n − B n ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) if γ < (cid:2) n − / { log log( n ) } γ { log( n ) } (1+ ε )( γ − (cid:3) if γ ≥ for arbitrary ε > . Proof of Theorem 2
According to Theorem 6 there exist series of Brownian bridges B n and B n such thatfor j = 1 , p ∈ [ δ nj , − δ nj ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q j ( p ) − Q j ( p ) q j ( p ) / √ n j − B n j ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n j − / log( n j ) (cid:9) . p ∈ [ δ n , − δ n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:115) s n n + s n (cid:40) (cid:98) Q ( p ) − Q ( p ) q ( p ) / √ n − B n ( y ) (cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:26)(cid:114) n n ( n + n ) log( n ) (cid:27) and sup p ∈ [ δ n , − δ n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:114) n n + s n (cid:40) (cid:98) Q ( p ) − Q ( p ) q ( p ) / √ n − B n ( p ) (cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:26)(cid:114) n n ( n + n ) log( n ) (cid:27) . The triangular inequality implies together with n = min { n , n } sup p ∈ [ δ n , − δ n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:114) n n n + s n (cid:40) s (cid:98) Q ( p ) − Q ( p ) q ( p ) − (cid:98) Q ( p ) − Q ( p ) q ( p ) (cid:41) − B n ,n ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a.s. = Ø (cid:8) n − / log( n ) (cid:9) , where B n ,n ( p ) = (cid:115) s n n + s n B n ( p ) − (cid:114) n n + s n B n ( p ) . By the independence of B and B it follows that B n ,n is a Brownian bridge as well. Theother parts of the theorem are proved in the same way. (cid:3) Proof of equation (5)
Assumption 4 states that q ( p ) = s q ( p ), which is equivalent to f {Q ( p ) } = s f {Q ( p ) } .Function f j {Q j ( p ) } is known as the density quantile function. This function is positive onits support [0 , Q j ( p ) = x implies α j = (cid:90) f j {Q j ( p ) } dp = (cid:90) ∞−∞ { f j ( x ) } dx, j = 1 , . Therefore, f {Q ( p ) } = s f {Q ( p ) } if and only if s = α /α . (cid:3) roof of Lemma 1 Following the proof of Theorem 2, it is easy to see that (cid:114) n n n + s n (cid:40) (cid:98) Q ( p ) − Q ( p ) q ( p ) /s + (cid:98) Q ( p ) − Q ( p ) q ( p ) (cid:41) . in D ∗ n ,n ( p ; s ) − D n ,n ( p ; s ) converges uniformly to a Brownian bridge. Applying the lawof iterated logarithm for weighted quantile processes (Theorem 1 and Remark 3 in Einmahland Mason, 1988) with weight function [ p (1 − p )] ν yields the lemma. (cid:3) A.2 Proofs of Section 3
Proof of Theorem 3
The result follows from the Consequence 4.1.2 on p. 34 of Cs¨org˝o (1983), Theorem 2and Lemma 1. (cid:3)
Proof of Theorem 4
From Corollary 1 in (Cs¨org˝o and R´ev´esz, 1984) we can get under Assumptions 1 and 2that sup p ∈ [ ε n , − ε n ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q j (cid:18) p + c α √ n j (cid:19) − Q j ( p ) − c α − B n j ( p ) (cid:12)(cid:12)(cid:12)(cid:12) a.s. = ø p (1)and sup p ∈ [ ε n , − ε n ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) Q j (cid:18) p − c α √ n j (cid:19) − Q j ( p ) + c α − B n j ( p ) (cid:12)(cid:12)(cid:12)(cid:12) a.s. = ø p (1)for j = 1 , ε n = n δ − / , δ ∈ (0 , / n ,n →∞ P (cid:40) (cid:98) Q (cid:18) p − c α √ n (cid:19) − (cid:98) Q (cid:18) p + c α √ n (cid:19) ≤ Q ( p ) − Q ( p ) ≤ (cid:98) Q (cid:18) p + c α √ n (cid:19) − (cid:98) Q (cid:18) p − c α √ n (cid:19) ; ε n ≤ p ≤ − ε n (cid:41) P (cid:40) sup p ∈ [0 , | B ,n ( p ) + B ,n ( p ) | ≤ √ c α (cid:41) . From the independence on Brownian bridges for j = 1 and j = 2 follows P (cid:40) sup p ∈ [0 , | B ,n ( p ) + B ,n ( p ) | ≤ √ c α (cid:41) = P (cid:40) sup p ∈ [0 , (cid:12)(cid:12)(cid:12) √ B ( p ) (cid:12)(cid:12)(cid:12) ≤ √ c α (cid:41) = P (cid:40) sup p ∈ [0 , | B ( p ) | ≤ c α (cid:41) for some Brownian bridge B . (cid:3)(cid:3)