A Log Probability Weighted Moment Estimator of Extreme Quantiles
AA Log Probability Weighted Moment Estimator ofExtreme Quantiles
Frederico Caeiro
Universidade Nova de Lisboa, FCT and CMA
Dora Prata Gomes
Universidade Nova de Lisboa, FCT and CMA
October 3, 2018
Abstract:
In this paper we consider the semi-parametric estimation of extreme quantiles of a rightheavy-tail model. We propose a new Log Probability Weighted Moment estimator for extreme quantiles,which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-orderregular variation condition on the tail, of the underlying distribution function, we deduce the non degenerateasymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimallevels. In addition, the performance of the estimators is illustrated through an application to real data.
Let us consider a set of n independent and identically distributed (i.i.d.), or possibly weakly depen-dent and stationary random variables (r.v.s), X , X , . . . , X n , with common distribution function(d.f.) F . We shall assume that F := 1 − F has a Pareto-type right tail, i.e., with the notation g ( x ) ∼ h ( x ) if and only if g ( x ) /h ( x ) →
1, as x → ∞ , F ( x ) ∼ ( x/C ) − /γ , x → ∞ , (1.1)with γ > C > U ( t ) := F ← (1 − /t ) = inf { x : F ( x ) ≥ − /t } , t > γ , i.e.,lim t →∞ U ( tx ) U ( t ) = x γ . (1.2)Consequentially, we are in the max-domain of attraction of the Extreme Value distribution EV γ ( x ) := (cid:40) exp( − (1 + γx ) − /γ ) , γx > γ (cid:54) = 0exp( − exp( − x )) , x ∈ R if γ = 0 . (1.3)1 a r X i v : . [ s t a t . M E ] J a n nd denote this by F ∈ D M ( EV γ ). The parameter γ is called the extreme value index (EVI), theprimary parameter in Statistics of Extremes.Suppose that we are interested in the estimation of a extreme quantile q p , a extreme valueexceeded with probability p = p n →
0, small. Since q p = F ← (1 − p ) ∼ Cp − γ , p →
0, for anyheavy tailed model under (1.1), we will also need to deal with the estimation of the shape and scaleparameters γ and C , respectively. Let X n − k : n ≤ . . . ≤ X n − n ≤ X n : n denote the sample of the k + 1 largest order statistics (o.s.) of the sample of size n , where X n − k : n is a intermediate o.s., i.e., k is a sequence of integers between 1 and n such that k → ∞ and k/n → , as n → ∞ . (1.4)The classic semi-parametric estimators of the parameters γ and C , introduced in Hill (1975), areˆ γ Hk,n := 1 k k (cid:88) i =1 (ln X n − i +1: n − ln X n − k : n ) , k = 1 , , . . . , n − , (1.5)and ˆ C Hk,n := X n − k : n (cid:18) kn (cid:19) ˆ γ Hk,n , k = 1 , , . . . , n − , (1.6)respectively. The EVI estimator in (1.5) is the well know Hill estimator, the average of the logexcesses over the high threshold X n − k : n . The classic semi-parametric extreme quantile estimatoris the Weissman-Hill estimator (Weissman, 1978) with functional expressionˆ W H k,n ( p ) := X n − k : n (cid:16) knp (cid:17) ˆ γ Hk,n , k = 1 , , . . . , n − . (1.7)Most classical semi-parametric estimators of parameters of the right tail usually exhibit the sametype of behaviour, illustrated in Figure 1: we have a high variance for high thresholds X n − k : n , i.e.,for small values of k and high bias for low thresholds, i.e., for large values of k . Consequently, themean squared error (MSE) has a very peaked pattern, making it difficult to determine the optimal k , defined as the value k where the MSE is minimal. For a detailed review on the subject see forinstance Gomes et al. (2008) and Beirlant et al. (2012).Apart from the classical EVI, scale and extreme quantile estimators in (1.5), (1.6) and (1.7),respectively, we shall introduce in Section 2 the corresponding Log Pareto Probability WeightedMoment estimators. In Section 3, we derive their non degenerate asymptotic behaviour and presentan asymptotic comparison of the estimators under study at their optimal levels. The probability weighted moments (PWM) method, introduced in Greenwood et al. (1979) isa generalization of the method of moments. The PWM of a r.v. X , are defined by M p,r,s :=2
50 100 150 200 2500.000.020.040.060.080.10 k ( BIAS ) VARMSE
Figure 1: Illustration of the Asymptotic Squared Bias, Variance and Mean squared error patterns,as function of k , of most classical semi-parametric estimators, for a sample of size n = 250. E ( X p ( F ( X )) r (1 − F ( X )) s ), with p, r, s ∈ R . When r = s = 0, M p, , are the usual non-central moments of order p . Hosking et al. (1987) advise the use of M ,r,s because the relationbetween parameters and moments is usually simpler than for the non-central moments. Also, if r and s are positive integers, F r (1 − F ) s can be written as a linear combination of powers of F or 1 − F and usually work with one of the moments a r := M , ,r = E ( X (1 − F ( X )) r ) or b r := M ,r, = E ( X ( F ( X )) r ). Given a sample size n , the unbiased estimators of a r and b r are,respectively, ˆ a r = 1 n n − r (cid:88) i =1 (cid:0) n − ir (cid:1)(cid:0) n − r (cid:1) X i : n , and ˆ b r = 1 n n (cid:88) i = r +1 (cid:0) i − r (cid:1)(cid:0) n − r (cid:1) X i : n . The first semi-parametric Pareto PWM (PPWM) estimators for heavy tailed models appeared inCaeiro and Gomes (2011a), for the estimation of the shape and scale parameters γ and C , and inCaeiro et al. (2012), for the estimation of extreme quantiles and tail probabilities. Since all thosePPWM estimators use the sample mean, they are only consistent if 0 < γ <
1. Caeiro and Gomes(2013) generalized the estimators in Caeiro and Gomes (2011a) with a class of PPWM estimators,consistent for 0 < γ < /r with r >
0. In order to remove the right-bounded support of theprevious PPWM estimators and have consistent estimators for every γ >
0, we shall next introducenew semi-parametric estimators based on the log-moments l r := E ((ln X )(1 − F ( X )) r ) . r , the unbiased estimator of l r is given byˆ l r = 1 n n − r (cid:88) i =1 (cid:0) n − ir (cid:1)(cid:0) n − r (cid:1) ln X i : n . For the strict Pareto model with d.f. F ( x ) = 1 − ( x/C ) − /γ , x > C > γ > l r = ln( C ) / (1 + r ) + γ/ (1 + r ) .To obtain the tail parameters estimators of γ and C of a underlying model with d.f. under(1.1), we need the followings results: • X n − k : n C ( n/k ) γ converges in probability to 1, for intermediate k ; • the conditional distribution of X | X > X n − k : n , is approximately Pareto with shape parameter γ and scale parameter C ( n/k ) γ .The PLPWM estimators of γ and C , based on the k largest observations, areˆ γ PLPWM k,n := 1 k k (cid:88) i =1 (cid:18) − i − k − (cid:19) ln X n − i +1: n , k = 2 , . . . , n, (2.1)and ˆ C PLPWM k,n := (cid:16) kn (cid:17) ˆ γ PLPWMk,n exp { D k,n } , k = 2 , . . . , n, (2.2)with D k,n := k (cid:80) ki =1 (cid:16) i − k − − (cid:17) ln X n − i +1: n . Notice that ˆ γ PLPWM k,n is a weighted average of the k largest observations, with the weights g i,k := (2 − i − k − ). Since g i,k = − g k − i +1 ,k , the weights areantisymmetric and their sum is zero. On the basis of the limit relation q p ∼ Cp − γ , p →
0, we shallalso consider the following quantile estimatorˆ Q PLPWM k,n ( p ) := (cid:16) knp (cid:17) ˆ γ PLPWMk,n exp { D k,n } , k = 2 , . . . , n, (2.3)valid for γ > In this section we derive several basic asymptotic results for the EVI estimators in (1.5) and (2.1)and for the quantiles estimators, ˆ W H k,n ( p ) and ˆ Q PLPWM k,n ( p ). Asymptotic results for the scale C -estimators are not presented but can be obtained with an analogous proof.To ensure the consistency of the EVI semi-parametric estimators, for all γ >
0, we need toassume that k is an intermediate sequence of integers, verifying (1.4). To study the asymptotic4ehaviour of the estimators, we need a second order regular variation condition with a parameter ρ ≤ U ( tx ) /U ( t ) to x γ in (1.2) and is given bylim t →∞ ln U ( tx ) − ln U ( t ) − γ ln xA ( t ) = x ρ − ρ ⇔ lim t →∞ U ( tx ) U ( t ) − x γ A ( t ) = x γ x ρ − ρ , (3.1)for all x >
0, with | A | a regular varying function with index ρ and x ρ − ρ = ln x if ρ = 0. Theorem 3.1.
Under the second order framework, in (3.1), and for intermediate k , i.e, when-ever (1.4) holds, the asymptotic distributional representation of ˆ γ • k,n , with • denoting either H or P LP W M , is given by ˆ γ • k,n d = γ + σ • Z • k √ k + b • A ( n/k )(1 + o p (1)) , (3.2) where d = denotes equality in distribution, Z • k is a standard normal r.v., b H = 11 − ρ , b PLPWM = 2(1 − ρ )(2 − ρ ) , σ H = γ and σ PLPWM = 2 √ γ. If we choose the intermediate level k such that √ k A ( n/k ) → λ ∈ R , then, √ k (ˆ γ • k,n − γ ) d → N ( λ b • , σ • ) . Proof.
For the Hill estimator, the proof can be found in de Haan and Peng (1998). For the PLPWMEVI-estimator, note that (cid:80) ki =1 (cid:16) − i − k − (cid:17) = 0 and consequentlyˆ γ PLPWM k,n = 1 k k (cid:88) i =1 (cid:18) − i − k − (cid:19) ln X n − i +1: n X n − k : n = 1 k k (cid:88) i =1 g i,k ln X n − i +1: n X n − k : n , k < n. We can write X d = U ( Y ) where Y is a standard Pareto r.v., with d.f. F Y ( y ) = 1 − /y , y > k is intermediate, we can apply equation (3.1) with t = Y n − k : n and x = Y n − i +1: n /Y n − k : n d = Y k − i +1: k , 1 ≤ i ≤ k , to obtainln X n − i +1: n X n − k : n d = γ ln Y k − i +1: k + Y ρk − i +1: k − ρ A ( Y n − k : n )(1 + o p (1)) . Then, since nY n − k : n /k p →
1, as n → ∞ ,ˆ γ PLPWM k,n d = 1 k k (cid:88) i =1 g i,k (cid:40) γE k − i +1: k + Y ρk − i +1: k − ρ A ( n/k )(1 + o p (1)) (cid:41) , where { E i } i ≥ , denotes a sequence of i.i.d. standard exponential r.v.’s. The distributional represen-tation of the EVI-estimator ˆ γ PLPWM k,n follows from the results for linear functions of ordinal statistics(David and Nagaraja, 2003), i.e., Z PLPWM k = √ kσ PLPWM k (cid:80) ki =1 ( g i,k E k − i +1: k −
1) is a standard normalr.v. and k (cid:80) ki =1 g i,k Y ρk − i +1: k − ρ converges in probability towards − ρ )(2 − ρ ) , as k → ∞ .The asymptotic normality of √ k (ˆ γ • k,n − γ ) follows straightforward from the representation in dis-tribution in (3.2). 5 emark 3.1. Notice that ˆ γ PLPWM k,n has a smaller asymptotic bias, but a larger asymptotic variancethan ˆ γ H k,n . A more precise comparison of the EVI-estimators will be dealt in Section 3.2. Remark 3.2.
For intermediate k such that √ k A ( n/k ) → λ , finite, as n → ∞ , the AsymptoticMean Squared Error (AMSE) of any semi-parametric EVI-estimator, with asymptotic distributionalrepresentation given by (3.2) , is AM SE ( (cid:98) γ • n,k ) := σ • k + b • A ( n/k ) , where Bias ∞ ( (cid:98) γ • n,k ) := b • A ( n/k ) and V ar ∞ ( (cid:98) γ • n,k ) := σ • /k . Let k • denote the level k , such that AM SE ( (cid:98) γ • n,k ) is minimal, i.e., k • ≡ k • ( n ) := arg min k AM SE ( (cid:98) γ • n,k ) . If A ( t ) = γβt ρ , β (cid:54) = 0 , ρ < which holds for most common heavy tailed models, like the Fr´echet, Burr, Generalized Pareto orStudent’s t, the optimal k -value for the EVI-estimation through ˆ γ • n,k is well approximated by k • = (cid:18) σ • n − ρ ( − ρ ) b • γ β (cid:19) − ρ . (3.3) Remark 3.3.
The estimation of the shape second-order parameter ρ can be done using the classesof estimators in Fraga Alves et al. (2003), Ciuperca and Mercadier (2010), Goegebeur et al. (2010)or Caeiro and Gomes (2012). Consistency of those estimators is achieved for intermediate k suchthat √ kA ( n/k ) → ∞ as n → ∞ . For the estimation of the scale second-order parameter β , formodels with A ( t ) = γβt ρ , β (cid:54) = 0 , ρ < , we refer the reader to the estimator in Gomes and Martins(2002). That estimator is consistent for intermediate k such that √ kA ( n/k ) → ∞ as n → ∞ andestimators of ρ such that ˆ ρ − ρ = o p (1 / ln n ) . Further details on the estimation of ( ρ , β ) can befound in Caeiro et al. (2009). For the extreme quantile estimators in (1.7) and (2.3), their asymptotic distributional represen-tations follows from the next, more general, Theorem.
Theorem 3.2.
Suppose that • denotes any EVI-estimator with distributional representation givenby (3.2) . Under the conditions of Theorem 3.1, if p = p n is a sequence of probabilities such that c n := k/ ( np ) → ∞ , ln c n = o ( √ k ) and √ kA ( n/k ) → λ ∈ R , as n → ∞ , then, √ k ln c n (cid:32) ˆ Q • k,n ( p ) q p − (cid:33) d = √ k ln c n (cid:32) ˆ W • k,n ( p ) q p − (cid:33) d = √ k (cid:0) ˆ γ • k,n − γ (cid:1) (1 + o p (1)) . (3.4) Proof.
Since q p = U (1 /p ), we can writeˆ W • k,n ( p ) q p = X n − k : n U ( n/k ) . U ( n/k ) U ( nc n /k ) ( c n ) ˆ γ • k,n . Using the second order framework, in (3.1), with t = n/k and x = kn Y n − k : n , results in X n − k : n U ( n/k ) d =1 + γ √ k B k + o p ( A ( n/k )) where B k := √ k (cid:0) kn Y n − k : n − (cid:1) is asymptotically a standard normal random6ariable. Using the results in de Haan and Ferreira (2006), Remark B.3.15 (p. 397), (cid:16) U ( c n .n/k ) U ( n/k ) c γn (cid:17) − =1 + A ( n/k ) ρ (1 + o (1)) follows. Then, since ( c n ) ˆ γ • k,n − γ d = 1 + ln( c n )(ˆ γ • k,n − γ )(1 + o p (1)), we getˆ W • k,n ( p ) q p d = 1 + ln( c n )(ˆ γ • k,n − γ )(1 + o p (1)) + γB k √ k + A ( n/k ) ρ (1 + o p (1)) , and the second equality in (3.4) follows immediately.For the other quantile estimator, we can writeˆ Q • k,n ( p ) = X n − k : n (cid:16) knp (cid:17) ˆ γ • k,n exp { ˜ D k,n } = ˆ W • k,n ( p ) exp { ˜ D k,n } , with ˜ D k,n := k (cid:80) ki =1 (cid:16) i − k − − (cid:17) ln X n − i +1: n X n − k : n . Then, since we haveexp { ˜ D k,n } d = 1 + γ √ k P k − ρA ( n/k )(1 + o p (1))(1 − ρ )(2 − ρ ) , with P k a standard normal r.v., the first equality in (3.4) follows. We now proceed to an asymptotic comparison of the PLPWM EVI estimator in (2.1) with the Hillestimator in (1.5) and the PPWM EVI estimator in Caeiro and Gomes (2011a), at their optimallevels. This comparison is done along the lines of de Haan and Peng (1998), Gomes and Martins(2001), Caeiro and Gomes (2011b), among others. Similar results hold for the extreme quantileestimators, at their optimal levels, since they have the same asymptotic behaviour as the EVIestimators, although with a slower convergence rate.Let k • be the optimal level for the estimation of γ through (cid:98) γ • k,n given by (3.3), i.e., the levelassociated with a minimum asymptotic mean square error, and let us denote (cid:98) γ • n := (cid:98) γ • k • ,n , theestimator computed at its optimal level. Dekkers and de Haan (1993) proved that, whenever b • (cid:54) = 0, there exists a function ϕ ( n ; γ, ρ ), dependent only on the underlying model, and not on theestimator, such thatlim n →∞ ϕ ( n ; γ, ρ ) AM SE ( (cid:98) γ • n ) = (cid:0) σ • (cid:1) − ρ − ρ (cid:0) b • (cid:1) − ρ =: LM SE ( (cid:98) γ • n ) . (3.5)It is then sensible to consider the following: Definition 3.1.
Given two biased estimators (cid:98) γ (1) n,k and (cid:98) γ (2) n,k , for which distributional representationsof the type (3.2) hold with constants ( σ , b ) and ( σ , b ) , b , b (cid:54) = 0 , respectively, both computed attheir optimal levels, k (1)0 and k (2)0 , the Asymptotic Root Efficiency ( AREF F ) indicator is definedas AREF F | := (cid:114) LM SE (cid:16)(cid:98) γ (2) n (cid:17) /LM SE (cid:16)(cid:98) γ (1) n (cid:17) = (cid:32)(cid:18) σ σ (cid:19) − ρ (cid:12)(cid:12)(cid:12)(cid:12) b b (cid:12)(cid:12)(cid:12)(cid:12)(cid:33) − ρ , (3.6) with LMSE given in (3.5) and (cid:98) γ ( i ) n := (cid:98) γ ( i ) k ( i )0 ,n , i = 1 , . emark 3.4. Note that this measure was devised so that the higher the AREFF indicator is, thebetter the first estimator is.
Remark 3.5.
For the PPWM EVI estimator, in Caeiro and Gomes (2011a), we have b PPWM = (1 − γ )(2 − γ )(1 − γ − ρ )(2 − γ − ρ ) and σ PPWM = γ √ − γ (2 − γ ) √ − γ √ − γ , < γ < . . To measure the performance of ˆ γ PLPWM k,n , we have computed the AREFF-indicator, in (3.6), asfunction of the second order parameter ρ . In figure 2 (left), we present the values of AREF F
P LP W M | H ( ρ ) = (cid:32)(cid:18) (cid:19) − ρ (cid:16) − ρ (cid:17)(cid:33) − ρ , (3.7)as a function of ρ . This indicator has a maximum near ρ = − .
7, and we have
AREF F
P LP W M | H >
1, if − . < ρ <
0, an important region of ρ values in practical applications. It is also easy tocheck that lim ρ →−∞ AREF F
P LP W M | H ( ρ ) = √ / ≈ .
866 and lim ρ → AREF F
P LP W M | H ( ρ ) = 1.In figure 2 (right) we show a contour plot with the comparative behaviour, at optimal levels, ofthe PLPWM and PPWM EVI-estimators in an important region of the ( γ , ρ )-plane. The greycolour marks the area where AREF F
P LP W M | P P W M >
1. At optimal levels, there is only asmall region of the ( γ , ρ )-plane where the AREFF indicator is slightly smaller than 1. Also, the AREF F
P LP W M | P P W M indicator increases, as γ increases and/or ρ decreases. −5 −4 −3 −2 −1 00.900.951.001.051.10 AREFF
PLPWM H r A R E FF AREFF
PLPWM PPWM g r . . . . . . . Figure 2:
Left:
Plot with the indicator
AREF F
P LP W M | H ( ρ ), in (3.7), as a function of ρ . Right:
Contour plot with the indicator
AREF F
P LP W M | P W M , as a function of ( γ , ρ ).8 A Case Study
As an illustration of the performance of the estimators under study, we shall next consider theanalysis of the Secura Belgian Re automobile claim amounts exceeding 1,200,000 Euro, over theperiod 1988-2001. This data set of size n = 371 was already studied by several authors (Beirlant et al. , 2004; Beirlant et al. , 2008 and Caeiro and Gomes, 2011c). g ^ H g ^ PLPWM H Q^ PLPWM
Figure 3:
Left:
Estimates of the EVI for the Secura Belgian Re data;
Right:
Estimates of thequantile q p with p = 0 .
001 for the Secura Belgian Re data.In Figure 3, we present, at the left, the EVI estimates provided by the Hill and PLPWMEVI-estimators in (1.5) and (2.1), respectively. At the right we present the corresponding quantileestimates provided by Weissman-Hill and PLPWM estimators, in (1.7) and (2.3), with p = 0 . k + 1 largest o.s.’s. For this dataset, we have ˆ ρ = − . β = 0 . k = [ n . ] = 368 (Caeiro and Gomes, 2011c). Usingthese values, the estimates of the optimal level, given by (3.3), are ˆ k H = 55 and ˆ k P LP W M = 76.Consequently, we have ˆ γ H , = 0 .
291 and ˆ γ PLPWM , = 0 . W H , ( p ) = 12622248 and ˆ Q P LP W M , ( p ) = 12373324. Based on the results here presented we can make the following comments: • Regarding efficiency at optimal levels, the new PLPWM estimators are a valid alternativeto the classic Hill, Weissman-Hill and PPWM estimators. And they are consistent for any γ >
0, which does not happen for the PPWM estimators.9
The analysis of the automobile claim amounts gave us the impression that the PLPWM EVIand extreme quantile estimators have a much smoother sample pattern than the Hill and theWeissman-Hill estimators. • It is also important to study the behaviour of the new PLPWM estimators for small samplesizes. That topic should be adressed in future research work. acknowledgement:
Research partially supported by FCT – Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia, project PEst-OE/MAT/UI0297/2011 (CMA/UNL), EXTREMA, PTDC/MAT /101736/2008.
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