Distribution sensitive estimators of the index of regular variation based on ratios of order statistics
DDistribution sensitive estimators of the index of regularvariation based on ratios of order statistics
Pavlina K. Jordanova and Milan Stehl´ık Faculty of Mathematics and Informatics, Konstantin Preslavsky University of Shumen,115 ”Universitetska” str., 9712 Shumen, Bulgaria. Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, Iowa, USA. Institute of Statistics, Universidad de Valpara´ıso, Valpara´ıso, Chile. Department of Applied Statistics, Johannes Kepler University, Altenbergerstrasse 69, 4040 Linz, Austria. a) Corresponding author: pavlina [email protected] b) [email protected] Abstract.
Ratios of central order statistics seem to be very useful for estimating the tail of the distributions and therefore, quantilesoutside the range of the data. In 1995 Isabel Fraga Alves investigated the rate of convergence of three semi-parametric estimatorsof the parameter of the tail index in case when the cumulative distribution function of the observed random variable belongsto the max-domain of attraction of a fixed Generalized Extreme Value Distribution. They are based on ratios of specific lineartransformations of two extreme order statistics. In 2019 we considered Pareto case and found two very simple and unbiasedestimators of the index of regular variation. Then, using the central order statistics we showed that these estimators have manygood properties. Then, we observed that although the assumptions are di ff erent, one of them is equivalent to one of Alves’sestimators. Using central order statistics we proved unbiasedness, asymptotic consistency, asymptotic normality and asymptotice ffi ciency. Here we use again central order statistics and a parametric approach and obtain distribution sensitive estimators of theindex of regular variation in some particular cases. Then, we find conditions which guarantee that these estimators are unbiased,consistent and asymptotically normal. The results are depicted via simulation study. INTRODUCTION AND PRELIMINARIES
Let us assume that X , X , ..., X n are n independent observations on a random variable (r.v.) X with cumulative distri-bution function (c.d.f.) F X ( x ) = P ( X ≤ x ) with regularly varying right tail. More precisely, we suppose that for some α >
0, lim t →∞ P ( X > tx ) P ( X > t ) = lim t →∞ ¯ F X ( tx )¯ F X ( t ) = x − α , ∀ x > , where ¯ F X ( x ) = − F X ( x ). Briefly we will denote these limit relations in this way ¯ F X ∈ RV − α . The distributions of orderstatistics X (1 , n ) ≤ X (2 , n ) ≤ ... ≤ X ( n , n ) are very well investigated in the scientific literature. One can see for exampleWilks (1948)[23], Renyi (1953)[21], Arnold (1992-2015)[3, 4], or Nevzorov (2001)[18].The task for estimation of the index of regular variation α > Q ∗ i , s , defined in (1). She works mainly with extreme order statistics. Here we consider central order statistics and thefollowing estimators Q i , s : = log X ( is , n ) X ( i , n ) H is − − H i − , Q ∗ i , s : = log X ( is , n ) X ( i , n ) log( s ) , (1) a r X i v : . [ m a t h . S T ] J u l LLi , s : = Q i , s , Q LL ∗ i , s = Q ∗ i , s , (2) Q Fr ∗ i , s : = − log X ( is , n ) X ( i , n ) log (cid:104) − log( s )log( s + (cid:105) , (3) Q HH ∗ i , s : = log X ( is , n ) X ( i , n ) + log (cid:104) − log( s )log( s + (cid:105) log( s ) , (4)where s = , , ... , n = i ( s + −
1, and i = , , ... . The fact that these estimators are functions only of ratios oforder statistics entails the invariance of these estimators with respect to a deterministic scale change of the sample.Therefore, without lost of generality, everywhere in this work we assume that the scale parameters σ in the considereddistributions are equal to 1.In 2019 Jordanova and Stehlik [17], assumed that the observed r.v. is Pareto distributed. Then, they have used thewell-known formulae for the mean and the variance of logarithmic di ff erences of the corresponding order statistics,and proved that Q i , s , and Q ∗ i , s estimators are unbiased (the second one only asymptotically), consistent, asymptoticallye ffi cient, and asymptotic normal. Due to the fact that we fix the exact probability type of the observed r.v. they donot impose separately the second order regularly varying condition defined in Geluk et al. (1997) [11]. For Paretodistribution it is automatically satisfied. Here we follow the same approach for di ff erent probability types. Its mainadvantage is that it is very flexible and provides an useful accuracy given mid-range and small samples. Log-Logistic,Fr´ e chet and Hill-horror cases are partially investigated. The conducted simulation study depicts the quality of theresults.Further on we denote by H n , m = + m + m + ... + n − m + n m , n = , , ..., the n -th Generalised harmonic number of power m = , , ... and H n : = H n , , denotes the n -th harmonic number.Along the paper we denote by F ← ( p ) = inf { x ∈ R : F ( x ) ≥ p } , the theoretical left-continuous version of thequantile function of a c.d.f. F , for p ∈ (0 ,
1] and by assumption F ← (0) : = sup { x ∈ R : F ( x ) = } , sup ∅ = −∞ . Thefollowing definition of empirical quantile function F ← n (0) : = X (1 , n ) , F ← n ( p ) = (cid:40) X ([ np + , n ) , np (cid:60) N , X ( np , n ) , np ∈ N , = (cid:40) X ( (cid:100) np (cid:101) , n ) , np (cid:60) N , X ( np , n ) , np ∈ N , p ∈ (0; 1] , (5)where [ a ] means the integer part of a and (cid:100) a (cid:101) is for the ceiling of a , i.e. the least integer greater than or equal to a ,could be seen e.g. in Serfling (2009) [22]. It is equivalent to the Definition 1, in Hyndman and Fan (1996) [15] and isimplemented in function quantile in software R [20], with parameter T ype = LOG-LOGISTIC CASE
In this section we consider a sample of n independent observations on a r.v. X with c.d.f. F X ( x ) = + x − α , x > . (6)Briefly X ∈ Log − Logistic ( α ; 0 , Q LLi , s and Q LL ∗ i , s and investigate their properties. The idea comes from thefollowing considerations. By formula (6) one can see that X α ∈ Log − Logistic (1; 0 , α log( X )is standard Logistic distributed, i.e. its location parameter is 0, its scale and shape parameters are equal to 1. The lastconclusion allows us to reduce the task for estimation of the parameter α , to the one of investigation of the propertiesf order statistics in the Logistic case which are already very well investigated in the scientific literature. In 1963Birnbaum and Dudman[6] found their moment and cumulant generating functions. Then, by using their derivativesthe authors expressed the mean and the variance of Logistic order statistics via polygamma function correspondinglyof the first and the second order. Gupta and Balakrishnan (1991) [12] made a step further on and found a seriesrepresentation of the joint moments of these order statistics. Their closed form solution seems to be still an openproblem.The following result is an immediate corollary of Theorem 4.6,a), in Jordanova (2020) [16]. Theorem 1.
Let us fix s = , , ... and consider a sample of i ( s + − i ∈ N independent observations on a r.v. X ∈ Log − Logistic ( α ; 0 , α >
0. Then, for x > α Q LL ∗ i , s is f α Q LL ∗ i , s ( x ) = s ) [ i ( s + − s − ix (1 − s − x ) i ( s − − [( i − [ i ( s − − (cid:90) ∞ z is − (1 + z ) is (1 + zs − x ) is dz , = s ) [ i ( s + − is − s xi ( s − (1 − s − x ) i ( s − − [ i ( s − − i − (2 is − F ( is , is ; 2 is ; 1 − s x )where F ( a , b ; c ; y ) = (cid:80) ∞ n = a ) n ( b ) n ( c ) n y n n ! = (cid:82) x b − (1 − x ) c − b − (1 − yx ) a B ( b , c − b ) dx , ( q ) n = q ( q + ... ( q + n − q ) = c > b > y <
1, isthe Gauss hypergeometric function (the Euler type integral) and f α Q LL ∗ i , s ( x ) =
0, otherwise.When compute the quantile function of the r.v. X with c.d.f. (6) we observe that for all s ∈ N ,log F ← X ( ss + ) F ← X ( s + ) = α log( s ) . In Jordanova and Stehl´ık (2019), in general (not only for Log-Logistic) case we have shown that,log (cid:32) X ( is , i ( s + − X ( i , i ( s + − (cid:33) P → i →∞ log F ← X ( ss + ) F ← X ( s + ) , s ∈ N (7)Therefore, in this case log (cid:32) X ( is , i ( s + − X ( i , i ( s + − (cid:33) P → i →∞ α log( s ) , s ∈ N (8)and when we normalize log (cid:16) X ( is , i ( s + − X ( i , i ( s + − (cid:17) with 2 log( s ) we obtain a consistent estimator for α . The last one is exactly Q LL ∗ i , s estimator, defined in (3). In order to obtain Q LLi , s estimator we normalize log (cid:16) X ( is , i ( s + − X ( i , i ( s + − (cid:17) with its expectation. Theproof of the following theorem could be found in Jordanova (2020) [16]. The approach is analogous to the one usedin Jordanova and Stehl´ık (2019) [17] for estimation of the parameter α in Pareto case. Theorem 2.
Let us fix s = , , ... and consider a sample of i ( s + − i ∈ N independent observations on a r.v. X ∈ Log − Logistic ( α ; 0 , α > i) For all i = , , ... , E [ Q LLi , s ] = α . ii) Q LLi , s a . s . → i →∞ α . iii) E ( Q LL ∗ i , s ) = H is − − H i − α log( s ) → i →∞ α . iv) Q LL ∗ i , s a . s . → i →∞ α . v) √ i ( s + − H is − − H i − ) (cid:104) α Q LLi , s − log( s ) H is − − H i − (cid:105) d → i →∞ η, where η ∈ N (cid:16) , s + ( s − s (cid:17) . vi) √ i ( s + − (cid:104) α Q LL ∗ i , s − (cid:105) d → i →∞ η, where η ∈ N (cid:16) , ( s + ( s − s [log( s )] (cid:17) . Remarks.
The first statement in Theorem 2 means that for all i ∈ N and s = , , ... , Q LLi , s are unbiased estimatorsfor α . The limit relations ii) and iv) say that both Q LLi , s and Q LL ∗ i , s are a strongly consistent when i → ∞ . The thirdone states that Q LL ∗ i , s is asymptotically unbiased estimator for α . Points v) and vi) clarify that both Q LLi , s and Q LL ∗ i , s areasymptotically normal when i increases unboundedly.he limit relation vi) in Theorem 2 allows us to obtain large sample confidence intervals for α . It means that s log( s ) s + (cid:115) i ( s + − s − (cid:16) α Q LL ∗ i , s − (cid:17) d → i →∞ θ, θ ∈ N (0 , . Thus, if we choose confidence level (1 − α )100%, where α ∈ (0 ,
1) and if we denote by z − α the 1 − α quantile ofthe Standard Normal distribution, thenlim i →∞ P − z − α ≤ s log( s ) s + (cid:114) i ( s + − s − (cid:16) α Q LL ∗ i , s − (cid:17) ≤ z − α = − α . Therefore,lim i →∞ P Q LL ∗ i , s − z − α ( s + Q LL ∗ i , s s log( s ) (cid:115) s − i ( s + − ≤ α ≤ Q LL ∗ i , s + z − α ( s + Q LL ∗ i , s s log( s ) (cid:115) s − i ( s + − = − α . Now Jordanova (2020) [16] uses the definition of Q LL ∗ i , s and concludes that for any fixed s = , , ... , and i ∈ N large enough, the corresponding 1 − α -confidence intervals (c.is.) for α are: s )log X ( is , i ( s + − X ( i , i ( s + − − z − α ( s + s log X ( is , i ( s + − X ( i , i ( s + − (cid:115) s − i ( s + −
1] ; 2 log( s )log X ( is , i ( s + − X ( i , i ( s + − + z − α ( s + s log X ( is , i ( s + − X ( i , i ( s + − (cid:115) s − s [ i ( s + − . (9)The task for estimation of the quantiles outside the range of the data seems to be more di ffi cult. It is easy to seethat if p < n , then by using the definition (5) we will obtain one and the same estimator for F ← X (1 − p ) and it is X ( n , n ) .In order to improve it we use formula (6) and more precisely its inverse function F ← X ( p ) = (cid:16) p − p (cid:17) /α , and obtain thefollowing two estimators ˆ F ← i , s , LL ( p ) = (cid:32) p − p (cid:33) Q LLi , s , (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) = (cid:32) p − p (cid:33) Q LL ∗ i , s . (10)Although they are not asymptotically normal, the following theorem explains why they are better than X ( n , n ) forestimation for 1 − p -th quantile, F ← X (1 − p ), p < n , which is outside the range of the data. Its statements follow byformulae (10), Theorem 2, ii) and iv), continuity of the function g ( x ) = (cid:16) p − p (cid:17) x and Continuous mappings theorem. Theorem 3.
Let us fix s = , , ... and consider a sample of n = i ( s + − i ∈ N independent observations on ar.v. X ∈ Log − Logistic ( α ; 0 , α >
0. Then i) ˆ F ← i , s , LL ( p ) a . s . → i →∞ F ← X ( p ). ii) (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) a . s . → i →∞ F ← X ( p ). Simulation study
The rate of convergence of Q LL ∗ i , s to α is quite good and it is partially depicted in Jordanova (2020) [16]. For α < p = given small samples (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) estimators are quite rough, therefore, here we have skippedtheir plots. The plots in Figure 1 depict the rate of convergence of (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) to F ← X ( p ) for di ff erent s = s = s = s = α = . α =
2. The estimated value of F ← X ( p ) is presented via a straight solid line. In order to plot these figures for di ff erent but fixed values of s and α , wehave simulated 1000 samples of n = s + − X ∈ Log − Logistic ( α ; 0 , i = , , ...,
150 we have computed (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ), p = , based onformula (10). Finally, we have averaged (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) over these 1000 samples and we have plotted these averages as afunction of i = , , ..., n = i ( s + −
1. These are i and s . We observe that when the sample size increases the estimators (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) get closer to the estimated value F ← X ( p ).These harmonize with our results in Theorem 3, ii) which says that (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) is a strongly consistent estimator for F ← X ( p ). Here the sample size n <
899 and p = / (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) is an estimator of a quantile outside the range of the data. IGURE 1.
Dependence of (cid:16) ˆ F ← i , s , LL (cid:17) ∗ ( p ) on i in Log-Logistic case. FR ´ E CHET CASE
Let us now assume that the observed r.v. has c.d.f. F X ( x ) = x ≤ x > F X ( x ) = exp {− x − α } . Brieflywe will denote this by X ∈ Fr ´ echet ( α, , X α ∈ Fr ´ echet (1 , α log( X ) ∈ Gumbel (0 , α we have to estimate it together with thescale and the location parameters. The corresponding MLE system of equations has no closed form solution. Prescottand Walden (1980)[19] proposed the last approach under more general settings, and more precisely for the familyof all Generalized Extreme Values(GEV) distributions. The Probability of Waited Moments system of equations isderived by Hosking et al. (1985) [14]. The authors apply numerical methods in order to obtain its solution. In 2015de Haan and Ferreira [10], suppose that the c.d.f. of the observed r.v. belongs to the max-domain of attraction ofsome GEV distribution and investigate the Block maxima estimator for the shape parameter. The corresponding MLEasymptotic theory was recently developed by Dombry and Ferreira [8]. Due to the generality of their assumptions,however, their method is applicable only in cases of huge samples. B¨ucher and Segers (2018) [7] generalize the resultsto stationary time series data, with distribution which belongs to the max-domain of attraction of Fr´ e chet distribution.They describe the disadvantages of Peaks-over-threshold and Block maxima methods. The authors point out thatthese methods ”...arise from asymptotic theory and are not necessarily accurate at sub-asymptotic thresholds or atfinite block lengths.” Here we propose a simple parametric estimator Q Fr ∗ i , s , defined in (3) for the parameter α . It isinvariant with respect to a scale change of the sample. The proof of the following theorem could be seen in Jordanova(2020) [16]. Theorem 4.
Let us consider a sample of n independent observations of a r.v. X ∈ Fr´ e chet( α , 0, 1), α > i) For any s = , , ... , i = , , ... , and y > f α Q Fr ∗ i , s ( y ) = c s C i , s e − yc s (cid:90) ∞ z exp( − iz ) (cid:8) exp[ − ze − yc s ] − exp( − z ) (cid:9) i ( s − − (cid:8) − exp[ − ze − yc s ] (cid:9) i − exp[ − ze − yc s ] dz , where c s = − log (cid:104) − log( s )log( s + (cid:105) and C i , s = [ i ( s + − i − [ i ( s − − . If y ≤ f α Q Fr ∗ i , s ( y ) = ii) Q Fr ∗ i , s estimator is strongly consistent, i.e. Q Fr ∗ i , s a . s . → i →∞ α .In order to estimate quantiles outside the range of the data in this case we use the following estimator (cid:16) ˆ F ← i , s , Fr (cid:17) ∗ ( p ) = [ − log( p )] − Q Fr ∗ i , s . (11) IGURE 2.
Dependence of (cid:16) ˆ F ← i , s , Fr (cid:17) ∗ ( p ) on i in Fr´ e chet case. The proof of the following theorem is analogous to the one of Theorem 3.
Theorem 5.
Let us fix s = , , ... and consider a sample of n = i ( s + − i ∈ N independent observations on ar.v. X ∈ Fr´ e chet( α , 0, 1), α >
0. Then (cid:16) ˆ F ← i , s , Fr (cid:17) ∗ ( p ) a . s . → i →∞ F ← X ( p ). Simulation study
The limit behaviour of Q Fr ∗ i , s is investigated and partially depicted in Jordanova (2020) [16]. Along the currentstudy we observed that for α < p = given small samples again (cid:16) ˆ F ← i , s , Fr (cid:17) ∗ ( p ) estimators are applicable onlyfor very large samples. Analogously to the previous case we have plotted the images in Figure 2 for α = . α = (cid:16) ˆ F ← i , s , Fr (cid:17) ∗ ( p ) for s = s = s = s = F ← X ( p ). We observe again the strong consistency of the estimators and can conclude that they havevery similar properties of the corresponding estimators considered in the Log-Logistic case in the previous section. HILL-HORROR CASE
Let α >
0. Embrechts et al. (2013) [9] define the following distribution via its quantile function, F ← X ( p ) = − log(1 − p )(1 − p ) /α , p ∈ (0 , X ∈ HH ( α ). This distribution is called Hill-horror distribution with parameter α , because of the di ffi culties related with the estimation of its parameter α via Hill[13] estimator. In this case, thedistribution of α log( X ) depends on α . Therefore, if we consider logarithms of ratios of two order statistics and computetheir means (if they exist) the result will depend non-linearly on α . In order to obtain a strongly consistent estimator of α , before we normalise the logarithm of fractions of order statistics with log( s ), we have shifted its distribution withan appropriate constant. It is determined by the equalitylog F ← X (cid:16) ss + (cid:17) F ← X (cid:16) s + (cid:17) = α log( s ) − log (cid:34) − log( s )log( s + (cid:35) . In this way we obtain Q HH ∗ i , s estimator, defined by formula (4). When we would like to estimate the quantiles outsidethe range of the data we obtain (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) = − log(1 − p )(1 − p ) QHH ∗ i , s . Now, the strong consistency of these estimators follows bya Continuous mappings theorem. IGURE 3.
Dependence of (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) on i in Hill-horror case. Theorem 6.
Let us fix s = , , ... and consider a sample of n = i ( s + − i ∈ N independent observations on ar.v. X ∈ HH ( α ), α >
0. Then, i) Q HH ∗ i , s a . s . → i →∞ α ; ii) (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) a . s . → i →∞ F ← X ( p ). Simulation study
Jordanova (2020) [16] compares this estimator with Hill[13] estimators and shows that Q HH ∗ i , s estimator has a relatively fast rate of convergence. In this section we will see that although ¯ F X ∈ RV − α and the statistic (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) is again a strongly consistent estimator of F ← X ( p ), its rate of convergence to the quantiles outside therange of the data is very slow. Our simulation study shows very di ff erent results than in the previous two cases. Theseconsiderations speak once again that although we have fixed α , the class of c.d.f. with regularly varying right tails withthis parameter α is too wide in order to be possible the parameter α to be simultaneously non-parametrically estimatedwithin this class. In order to plot our parametric estimators (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) we have followed a similar procedure fordrawing the graphs that was used in the previous two sections. We have simulated 1000 samples on n = s + − X ∈ HH ( α ) separately for α = . α =
2. For any fixed α , p = and s = , , , i = , , ...,
150 we have computed (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) estimators. Then, for any fixed s and i and over these1000 samples we have determined the average of (cid:16) ˆ F ← i , s , HH (cid:17) ∗ ( p ) estimators. Finally, we have plotted them in Figure3. Again we observe the strong consistency of the considered estimators, however the rate of convergence is muchslower than in previous two cases. This is very important especially when α approaches 0. The last means that theselast estimators are appropriate only if we are working with large samples. The sample size n = s + −
1, where s = , , ,
5, is not enough to observe their good properties.An analogous approach could be applied to many di ff erent cases of distributions with regularly varying righttails. It leads us to strongly consistent and distribution sensitive estimators of the index of regular variation. Thesecould be for example Exponentiated-Fr´ e chet, Burr, Reverse Burr, Danielson and de Vries, among others distributions.If the observed r.v. does not have c.d.f. with regularly varying tail, however it can be continuously transformed tosuch a distribution the same approach is applicable to the transformed r.vs. These are for example H , Log-Pareto, orWeibull distributions. CKNOWLEDGMENTS
The authors are grateful to the bilateral projects Bulgaria - Austria, 2016-2019, Feasible statistical modelling forextremes in ecology and finance, Contract number 01 /
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