Gini's mean difference and variance as measures of finite populations scales
aa r X i v : . [ m a t h . S T ] J un Gini’s mean difference and variance as measures offinite populations scales
Andrius ˇCiginas and Dalius Pumputis Vilnius University Institute of Mathematics and Informatics, LT-08663 Vilnius, Lithuania Lithuanian University of Educational Sciences, LT-08106 Vilnius, Lithuania
Abstract
We consider Gini’s mean difference statistic as an alternative to the empirical variance in the set-tings of finite populations where simple random samples are drawn without replacement. In particular,we discuss specific (in the finite population context) estimation strategies for a scale of the population,related to the alternative statistic under possible presence of outliers in the data.The paper presents also a wide comparative survey of properties of the Gini mean differencestatistic and the empirical variance. It includes asymptotic properties of both statistics: the asymptoticnormality, one-term Edgeworth expansions and bootstrap approximations for Studentized versions ofthe statistics. An estimation of the variances and other parameters of the statistics is also in the study,where we exploit an auxiliary information on the population elements in the case of its availability.Theoretical results are illustrated with a simulation study.
Keywords: sampling without replacement, sample variance, Gini’s mean difference, robustness, asymp-totic normality, second-order approximations
MSC classes:
Together with a location parameter, a spread (or scale) of a survey population are usually the parame-ters of interest. If a statistician assumes the classical model of independent and identically distributed(i.i.d.) observations, then, at least, he has at his disposal the number of parametric distributions families,e.g., Gaussian, Cauchy, etc. Assume that, he chooses the particular family for a further analysis of thedata. This family comes with its own measures of location and scale, for instance, the normal distributionparameters ‘suggest’ to measure the mean and variance of the survey population, and the Cauchy distribu-tion is specified by the population median and interquartile range. The traditional statistics theory has theanswers how to get efficient estimates of locations and scales under commonly used populations models.However, parametric statistics models, being comparatively convenient, are known also as non-robust,i.e., deviations from their assumptions may lead to misleading conclusions. As it is often an instance, an
The research of the first author is supported by European Union Structural Funds project ”Postdoctoral Fellowship Imple-mentation in Lithuania”. t , . . . , N u of elements with the corresponding set of real values X “ t x , . . . , x N u of the variable x under investigation, and for the simple random sample t , . . . , n u of size n ă N drawn without replace-ment from the population with the measurements X “ t X , . . . , X n u of the variable x . In particular, thepopulation parameters G “ ˆ N ˙ ´ ÿ ď i ă j ď N | x i ´ x j | (1.1)and V “ ˆ N ˙ ´ ÿ ď i ă j ď N p x i ´ x j q { X , only the latter seems more natural because of Var X “p N ´ q V { N . The corresponding unbiased estimators of these parameters are the GMD statistic U G “ ˆ n ˙ ´ ÿ ď i ă j ď n | X i ´ X j | (1.3)and the empirical variance U V “ ˆ n ˙ ´ ÿ ď i ă j ď n p X i ´ X j q { . (1.4)As an alternative to (1.4), the GMD statistic, known better since Gini (1912), is widely used in economics.Now it is an ordinary measure of a dispersion of a distribution of income and also in cases of similar vari-ables, see monograph of Yitzhaki and Schechtman (2013), where, by words of the authors, the commonlyused variance-based analyses are ‘translated’ into Gini-based. A use of the GMD is not restricted withmeasurements of an economic inequality. As in problems of economists, where data deviate from thenormality, the parameter G and its estimator U G can be used as dispersion’s measures for many kinds ofstatistical data. Our choice of the finite populations setting has a motivation from the side of economicstoo, because, in economical surveys, the number N of surveyed objects or subjects is not necessarily solarge (compared to the sample size) that to ignore a dependence between the observations in the set X .In Section 2, we consider three estimation of the finite population scale strategies related to the alter-native U G . We exploit two assumptions, which are usually possible in the finite population context: theso-called superpopulation assumption, and the availability of an auxiliary information about the popula-tion elements. We perform also simulation experiments, where we analyze advantages and disadvantagesof the strategies, and compare them under populations without and with outliers.2he GMD statistic is one of several well-known universal estimators as, e.g., the median absolutedeviation, interquartile range, which are less sensitive to outliers than the sample variance. Looking fromthe side of the robust estimation theory, if we can link data to a parametric population model, then, inthe particular situations, there are more effective robust estimators of scale than those common ones, seeHuber (1981). But we focus here on an unified improvement of the empirical variance.The next premium, which should be paid, is a relatively complex access to properties of the GMDstatistic. On the other hand, in these problems, U G is more attractive than the other mentioned examplesof universal estimators because of its smoothness (in a certain sense) or that it uses the complete sampleinformation. In Section 3, an asymptotic analysis of distributions of the statistics U G and U V shows thattheir properties are similarly simple. To explain it, we apply an available theory of U - and L -statisticsin the case of samples without replacement. In particular, statistics (1.3) and (1.4) are likely the mostpopular U -statistics of degree two, and (1.3) is also the L -statistic, see Serfling (1980). As the L -statistic, U G is smooth in the sense that its weight function is smooth, see ibidem.To be consistent with already known results, first, we mark that expressions of the variance of U G andits approximations are known since Nair (1936) and Lomnicki (1952) in the case of i.i.d. observations,and since Glasser (1962) for the simple random samples without replacement. Second, a strong method tostudy the variances and the asymptotic normality of the statistics U G and U V is Hoeffding’s decompositionfor U -statistics in Hoeffding (1948). Much latter, in Zhao and Chen (1990), the analogous decompositionwas used in the case of finite population. Third, similarly, second-order approximations theory for sam-ples without replacement has been realized after the case of i.i.d. observations: Kokic and Weber (1990),and Bloznelis and G¨otze (1999) follow Bickel et al. (1986) on one-term Edgeworth approximations tothe distributions of standardized U -statistics; papers of Bloznelis (2003), and Bloznelis (2007), on anone-term Edgeworth expansion for Studentized U -statistics and bootstrap approximations, appeared afterHelmers (1991).Since the true values of variances of the statistics are almost always unknown, we prefer to considerthe asymptotic normality, one-term Edgeworth expansions and bootstrap approximations for Studentizedversions of the statistics U G and U V . A basis for such a study is the general theory in Bloznelis and G¨otze(2001), Bloznelis (2003), and Bloznelis (2007) (with without-replacement bootstrap of Booth et al. (1994)),where, to ensure a validity of the approximations, quite general smoothness conditions are imposed onparts of the Hoeffding decomposition of U -statistics. Theorems of Section 3 let to compare distributionalproperties of U G and U V much easy.A successful application of the one-term Edgeworth expansion requires to have good estimators (inthe sense of an asymptotic consistency or a small mean square error) of the expansion’s parameters. Inthe case of symmetric statistics (symmetric functions of observations) including U -statistics, jackknifetechniques are used to estimate these parameters, see Putter and van Zwet (1998), and Bloznelis (2001).In the separate cases of statistics, for example, for U G and U V , there are more ways to construct estima-tors of the Edgeworth expansions parameters, e.g., for L -statistics including U G , the bootstrap was usedin ˇCiginas (2013a) and, assuming that the auxiliary information is available, calibration methods wereapplied in Pumputis and ˇCiginas (2013). In Section 4, we propose simple and also efficient estimators ofthe parameters, without the auxiliary information and also using it. Similar estimators of the variancesof U G and U V are also considered. In Section 5, we discuss empirical Edgeworth expansions, based onthe estimators of the parameters, and bootstrap approximations. In Section 6, we compare the obtainedestimation results for both statistics of interest in the simulation study. Here we are interested also in a3ole of outliers in populations. Conclusions of the paper are given in Section 7. In the i.i.d. setup, for many common parametric models of populations, the sample variance is an efficientestimator under ideal or close to ideal conditions. But assume that some of the sample data differ sub-stantially from the other. Then the GMD statistic can be a better choice because it puts smaller weightson extreme observations thus lowering their impact on the estimation.In the finite population case, outliers are less influential too, when U G is applied. To see it, let us writeparameters (1.1) and (1.2) in the different form. Assume (here and further in the paper), without loss ofgenerality, that x ď ¨ ¨ ¨ ď x N , and denote D i “ x i ` ´ x i , i “ , . . . , N ´
1. Then, taking x j ´ x i “ ř j ´ k “ i D k ,one can obtain G “ N p N ´ q « N ´ ÿ i “ i p N ´ i q D i ` ÿ ď i ă j ď N ´ i j p N ´ i qp N ´ j q D i D j ff and V “ N p N ´ q « N ´ ÿ i “ i p N ´ i q D i ` ÿ ď i ă j ď N ´ i p N ´ j q D i D j ff . These expressions are connected via the formal transformation D i D j “ j p N ´ i q N p N ´ q D i D j , ď i ď j ď N ´ X , where D i “ x i ` ´ x i , i “ , . . . , N ´
1, which explains the assertion. We note that system of equations(2.1) has not a solution X “ t x , . . . , x N u except in cases of very simple X . Outliers model.
For the simple random samples without replacement, we assume an existence of so-called representative outliers. This notion was introduced in Chambers (1986). It means the assumptionsthat: outlying observations are not errors of a measurement; the unsampled population part should containoutliers too. If these assumptions do not hold, then, in sample surveys, the problem of outliers is treatedusually as a different from the estimation.More formally, denote by 0 ď p ď N the number of outliers in the population. Assume that thepopulation elements t i , . . . , i p u Ď t , . . . , N u belong to a different population, but this phenomenon is notknown while the sample X was not obtained. Then the corresponding values from x , . . . , x N are treated asoutliers. In the random sample X , the number of outliers is random and equals to the number of elementsin the set t i , . . . , i p u X t , . . . , n u .The proportion p { N of outliers can be restricted without a significant loss of generality. In particular,as it is pointed in Huber (1981), a part of gross errors (outliers) in samples usually is not larger than10%. An interesting note on this issue is given in Chhikara and Feiveson (1980): ” . . . it is reasonableto consider three potential outliers in a data set of observations, but it is unrealistic to expect outliers out of a data set of observations. In the latter case, the outlier detection problem becomes ne of discrimination between two or more classes of data. ”. Similarly, for finite populations, if a largeportion of outliers is expected in the population, they are neutralized typically (with a help of an auxiliaryinformation) by applying stratified sampling designs, i.e., collecting potential outliers into a separatestratum. Another but similar solution, in this case, is a postratification. Estimation strategies.
Specific for the finite population ways to apply the GMD statistic as the alternativeto the sample variance are the following.( S ) Assume that the fixed numbers x , . . . , x N are the realizations of i.i.d. random variables X ˚ , . . . , X ˚ N (superpopulation model) from a parametric family of distributions with the scale parameter which is anone-argument function of a Var X ˚ . Then the scale of X is treated as the same function of ? V , and theestimator of the argument ? V is taken to be of the form aU G , where a ą S ) Under the presence of well-correlated and completely known auxiliary variable z with the values Z “ t z , . . . , z N u in the population, the scale measure is ? V and its estimator is aU G with the correction a ą Z .( S ) The parameter G is itself treated as the scale of X , and the GMD statistic U G is its estimator.Case ( S ) is close to the parametric statistics. In the i.i.d. settings, the multipliers a , which ensure that aU G is the unbiased estimator of ? V , are known for commonly used parametric families: a “ ? p { a “ Z is available, then these theoretical a should not so much differ from the correspondingvalues obtained by case ( S ), where a ą aG “ ? V using Z instead of X . If the scatterof X can not be linked to a distributions family, e.g., it is a mixture of two unknown distributions, andthere is no other additional information, then we suggest strategy ( S ). We compare efficiencies of strategies ( S ) and ( S ) in respect of the common estimation by ? U V un-der presence of outliers. We consider two populations which values of the variable x are generatedrespectively from two different parametric families: the normal distributions N p µ , s q , and the gammadistributions G p k , q q with the shape k and scale q , where variance is equal to k q . In the case of gammadistribution, the correction a “ k ´ { p ´ I . p k ` , k qq ´ depends on k , where I t p u , v q is the regularizedincomplete Beta function. For each of these populations, we consecutively increase the part of outliersin the population as follows. Firstly, we select some particular population elements randomly withoutreplacement. Secondly, we replace their values by new generated from the same family of distributionsbut with different parameters, and we fix these values. In the next steps, the set of outlying elements isincreased by selecting from those which still not belong to the outliers.In particular, the distributions are: N p , q , and N p , q is for generation of outliers; G p , {? q (then a “ ? { G p , ? q is for outliers. We take N “ n “ p “ , , , , ,
100 outliers.The fixed values of the auxiliary information Z are generated by the linear regression z i “ ` x i ` e i ,where e i , i “ , . . . , N , are i.i.d. random variables from N p , J q . Since the set X is different for different p , collections Z are different too.To understand better a role of the auxiliary information in strategy ( S ), we simulate different cor-5elations r zx between Z and X . The correlation is controlled with the variance J in the linear model.Thus we choose the variance in order to have r zx “ . , . , . N p , q with outliers N p , q . Accuracy by 10 ˆ p BIAS p¨q , a MSE p¨qq . p { N ? U V ( S ) r zx “ .
9; ( S ) r zx “ .
7; ( S ) r zx “ .
5; ( S )0 . p´ . , . q p´ . , . q p´ . , . q p . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q Table 2: G p , {? q with outliers G p , ? q . Accuracy by 10 ˆ p BIAS p¨q , a MSE p¨qq . p { N ? U V ( S ) r zx “ .
9; ( S ) r zx “ .
7; ( S ) r zx “ .
5; ( S )0 . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q . p´ . , . q p´ . , . q p´ . , . q p´ . , . q p´ . , . q It is seen from Table 1 that strategy ( S ) improves the estimator ? U V where the proportion p { N issmaller. For p { N larger than 0 .
04, ( S ) becomes inefficient (by MSE p¨q ) because its bias is large, since thefixed correction a is to much approximate for the mix of the normal distributions. Strategy ( S ) is the bestunder strong correlation between x and z , because the estimation bias is well-corrected. The efficiency of( S ) decreases with the decrease of the correlation r zx .Table 2 shows similar results for the asymmetric gamma distributions. Here outliers affect the es-timators stronger because the distribution of outliers has larger mean (location) in addition. Therefore,strategies ( S ) and ( S ) are efficient for smaller proportions p { N than in Table 1.We conclude that strategies ( S ) and ( S ), and thus the GMD statistic, are efficient, in respect of ? U V ,if there is a small percent of outliers in the population. Moreover, there is no loss in the efficiency of thestrategies if there are no outliers in the population. The statistic U “ U n p X q “ ř ď i ă j ď n h p X i , X j q , where a function h : X ˆ X Ñ R satisfies h p x , y q “ h p y , x q ,is called U -statistic of degree two. For the cases of the GMD statistic U G and the sample variance U V , wehave h p X , X q “ ˆ n ˙ ´ | X ´ X | h p X , X q “ ˆ n ˙ ´ p X ´ X q { , respectively. Following Bloznelis (2003), the Hoeffding decomposition of the U -statistic is U “ E U ` U ` U , (3.1)where U “ ř ni “ g p X i q and U “ ř ď i ă j ď n g p X i , X j q are centered and uncorrelated linear and quadraticparts, respectively. Here, for 1 ď k ď N , g p x k q “ p n ´ q N ´ N ´ E p h p X , X q ´ E h p X , X q | X “ x k q and, for 1 ď k ‰ l ď N , g p x k , x l q “ h p x k , x l q ´ E h p X , X q ´ p n ´ q ´ p g p x k q ` g p x l qq . The so-called first- and second-order influence functions g p¨q and g p¨ , ¨q have usually a different impactto the variance of U -statistic. As in cases of any other linearization techniques, it is expected that thelinear part in (3.1) dominates against the remainder in the sense of variance size. In particular, we considerstructures of the variances of the statistics U G and U V by formula (2.6) in Bloznelis and G¨otze (2001): Var U “ n p N ´ n q N ´ s ` ˆ n ˙ˆ N ´ n ˙ˆ N ´ ˙ ´ s , (3.2)where it is denoted s “ E g p X q and s “ E g p X , X q . Let us elaborate the statistics of interest. GMD statistic.
To find the influence functions, we rewrite (1.3) into the alternative form U G “ ˆ n ˙ ´ n ÿ j “ p j ´ n ´ q X j : n , where X n ď ¨ ¨ ¨ ď X n : n are the order statistics of the observations X , and apply the Hoeffding decom-position results for L -statistics from ˇCiginas (2012). Denote a i “ p i ´ N q{ N , 1 ď i ď N ´
1. Then, for1 ď k ď N , g p x k q “ ´ n NN ´ N ´ ÿ i “ ˆ I t i ě k u ´ iN ˙ a i D i , where I t¨u is the indicator function, and, for 1 ď k ă l ď N , g p x k , x l q “ ´ n p n ´ q N ´ ÿ i “ f k , l p i q D i , where f k , l p i q “ $’&’% i p i ´ q{ A , if 1 ď i ă k , ´p i ´ qp N ´ i ´ q{ A , if k ď i ă l , p N ´ i ´ qp N ´ i q{ A , if l ď i ă N ,7ith A “ p N ´ qp N ´ q . Next, direct calculations give the expressions of variance decomposition (3.2)components: s “ n p N ´ q « N ´ ÿ i “ i p N ´ i q a i D i ` ÿ ď i ă j ď N ´ i p N ´ j q a i a j D i D j ff (3.3)and s “ n p n ´ q N p N ´ q p N ´ q « N ´ ÿ i “ i p i ´ qp N ´ i ´ qp N ´ i q D i ` ÿ ď i ă j ď N ´ i p i ´ qp N ´ j ´ qp N ´ j q D i D j ff . (3.4) Sample variance.
Denote the population moments b “ E X and µ k “ E p X ´ b q k , for k “ , . . . , ď k ď N , g p x k q “ n NN ´ “ p x k ´ b q ´ µ ‰ , (3.5)and, for 1 ď k ă l ď N , g p x k , x l q “ n p n ´ q " p x k ´ x l q ` N p N ´ qp N ´ q µ ´ NN ´ “ p x k ´ b q ` p x l ´ b q ‰* . (3.6)After strightforward calculations, we obtain the following formulas: s “ n ˆ NN ´ ˙ ` µ ´ µ ˘ (3.7)and s “ n p n ´ q N p N ´ qp N ´ q ˆ N ´ N ` N ´ µ ´ µ ˙ . (3.8)In fact, various expressions of Var U V are known in the literature. For a comparison, we mention just thatappeared in Irwin and Kendall (1944). Common inferences about statistics are based on knowledge of their distributions. If exact distributionscannot be accessed, then, for samples of a sufficiently large size, the normal approximation to distributionsis usually appropriate. Here, for the statistics under investigation, we give sufficient and simple conditionswhere the distribution function F nS p y q “ P t U ´ E U ď yS u (3.9)of the Studentized U -statistic is asymptotically normal as the sample size increases. Here S “ S p X q “ ´ ´ nN ¯ n ´ n n ÿ i “ ` U n ´ p X z X i q ´ s U ˘ , where s U “ n n ÿ i “ U n ´ p X z X i q , (3.10)8s the jackknife estimator of the variance for any U -statistic.In the finite populations asymptotics, the population size increases together with the sample size. Wedenote n ˚ “ min t n , N ´ n u , which tends to infinity as n does in the i.i.d. setup. Next, to be correct inthe formulation of asymptotic results, a sequence of values X r “ t x r , , . . . , x r , N r u in the populations, with N r Ñ 8 as r Ñ 8 , and a sequence of statistics U n r p X r q , where X r “ t X r , , . . . , X r , n r u is a sample drawnwithout replacement from X r , should be considered. Further, we omit the subscript r for these and otherquantities for notational simplicity.Denote t “ n p ´ n { N q for short. Erd˝os and R´enyi (1959), and H´ajek (1960) Lindeberg-type condi-tion: for every e ą s ´ E g p X q I t| g p X q| ą ets u “ o p q as n ˚ Ñ 8 , (3.11)imposed on the linear part of U -statistic, is necessary for the normality of asymptotically linear statisticsas the size n ˚ grows. This condition, together with moments conditions ensuring the asymptotic linearity,is sufficient for the satistics U G and U V by the following limit theorem. Theorem 1.
Assume that n ˚ Ñ 8 . Let (3.11) be satisfied. Assume that for all n ˚ : (i) for U G , E X ď C ă8 holds; (ii) for U V , E X ď C ă 8 holds. Then, for U G and U V , (3.9) tends to the standard normaldistribution function F p y q for every y P R , respectively.Proof. To be consistent with conditions imposed on symmetric (and thus U -) statistics in Bloznelis and G¨otze(2001), consider normalized versions of the statistics of interest: ? nU G and ? nU V . Then the variancesof linear parts from the decompositions of these statistics are bounded away from zero, and are finite ifthe corresponding conditions (i) and (ii) are satisfied. Therefore, in the case of U G , the normality prooffollows immediately from Theorem 1 in ˇCiginas (2013b) through Proposition 3 in Bloznelis and G¨otze(2001). In the case of U V , by Theorem 1 and Proposition 3 in Bloznelis and G¨otze (2001), it suffices toverify that the variance of quadratic part of ? nU V tends to zero as n ˚ Ñ 8 . If (ii) is satisfied, it followseasily from the explicit formulas above.
When the sample size is not a large, the normal approximation to (3.9) can be inaccurate. Then theone-term Edgeworth expansion H nS p y q “ F p y q ` ` ´ n { N ` p ´ n { N q y ˘ a ` ` y ` ˘ k t j p y q , (3.12)for Studentized U -statistics, constructed in Bloznelis (2003), can be an improvement. Here j p y q is thestandard normal density function, and a “ s ´ E g p X q and k “ s ´ t E g p X , X q g p X q g p X q are the population characteristics. Next, we give detailed expressions of these parameters for both statis-tics of interest. 9 MD statistic.
Routine but tedious combinatorial calculations give a “ ´ s ´ n p N ´ q « N ´ ÿ i “ i p N ´ i qp N ´ i q a i D i ` ÿ ď i ă j ď N ´ i p N ´ i qp N ´ j q a i a j D i D j ` ÿ ď i ă j ď N ´ i p N ´ j qp N ´ j q a i a j D i D j ` ÿ ď i ă j ă m ď N ´ i p N ´ j qp N ´ m q a i a j a m D i D j D m ff (3.13)and k “ ´ s ´ t n p n ´ q N p N ´ q p N ´ q N ´ ÿ i “ N ´ ÿ j “ N ´ ÿ m “ c i jm a j a m D i D j D m , (3.14)where c i jm “ $’’’’’’’’’&’’’’’’’’’% i p i ´ qp N ´ m qr N ´ j ´ ` N ´ j p m ´ j qs , if i ď j ď m , i p i ´ qp N ´ j qr N ´ m ´ ` N ´ m p m ´ j qs , if i ď m ă j , j p N ´ m qrp i ´ qp N ´ i ´ q ` N ´ tp N ´ i qp N ´ i ´ qp i ´ j q ` i p i ´ qp m ´ i qus , if j ă i ă m , m p N ´ j qrp i ´ qp N ´ i ´ q ` N ´ t i p i ´ qp i ´ j q ` p N ´ i ´ qp N ´ i qp m ´ i qus , if m ă i ă j , j p N ´ i ´ qp N ´ i qr m ´ ` N ´ p N ´ m qp m ´ j qs , if j ă m ď i , m p N ´ i ´ qp N ´ i qr j ´ ` N ´ p N ´ j qp m ´ j qs , if m ď j ď i .These formulas are new in the literature. Sample variance.
With strightforward calculations one can arrive to the following results: a “ s ´ n ˆ NN ´ ˙ ` µ ´ µ µ ` µ ˘ (3.15)and k “ s ´ t n p n ´ q ˆ NN ´ ˙ N ´ ˆ ´p N ´ q µ ´ N ´ N ´ µ µ ` NN ´ µ ` µ ˙ . (3.16)Note that (3.16) can be simplified (approximated) by leaving the term with µ in the brackets only. Forcomparison, expressions similar to these can be identified in the Edgeworth approximation given byKokic and Weber (1990) for standardized sample variance.While an error of the normal approximation is typically of the order O p n ´ { ˚ q , see, e.g., Zhao and Chen(1990) for the case of standardized U -statistics, the error of the true (with known parameters a and k )one-term Edgeworth approximation (3.12) is of the order o p n ´ { ˚ q under certain conditions. The firstcondition, from those, is the asymptotical nonlatticeness of the linear part of U -statistic: for every e ą B ą
0, lim inf n ˚ Ñ8 sup e ă| t |ă B ˇˇˇ E exp ! i t s ´ g p X q )ˇˇˇ ă , (3.17)see Bloznelis and G¨otze (2001). This and other specific sufficient conditions for the statistics U G and U V are summarized in the following theorem. 10 heorem 2. Assume that n ˚ Ñ 8 and p ´ n { N q t Ñ 8 . Let (3.17) be satisfied. Assume that, for some d ą and for all n ˚ : (i) for U G , E | X | ` d ď C ă 8 holds; (ii) for U V , E | X | ` d ď C ă 8 holds. Then,we have sup y P R | F nS p y q ´ H nS p y q| “ o p n ´ { ˚ q as n ˚ Ñ 8 , for U G and U V , respectively.Proof. In the case of U G , the proof is the corollary of Theorem 1 in Bloznelis (2003) following techniquein the proof of Theorem 1 in ˇCiginas (2012). In particular, by these theorems, the boundedness of thecharacteristics b s “ s ´ s E | g p X q| s and g s “ s ´ s t s E | g p X , X q| s , as n ˚ Ñ 8 , must be verified for s ą U V , the task is the same. By (3.5), for s ě
1, applying inequalities | a ´ b | s ď s ´ p a s ` b s q where a , b ě
0, and µ s ď µ s , we get E | g p X q| s “ N N ÿ k “ | g p x k q| s ď s ´ n s ˆ NN ´ ˙ s N N ÿ k “ ` p x k ´ b q s ` µ s ˘ ď ˆ Nn p N ´ q ˙ s µ s . (3.18)By (3.6), for 1 ď k ă l ď N , applying p x k ´ x l q ď ` p x k ´ b q ` p x l ´ b q ˘ , we have | g p x k , x l q| ď n p n ´ q ˆ N ´ N ´ ` p x k ´ b q ` p x l ´ b q ˘ ` N p N ´ qp N ´ q µ ˙ ď n p n ´ q NN ´ ` p x k ´ b q ` p x l ´ b q ` µ ˘ . Then, for s ě
1, similarly as in (3.18), applying p a ` b q s ď s ´ p a s ` b s q twice, where a , b ě
0, and notingthat ř ď k ă l ď N ` p x k ´ b q s ` p x l ´ b q s ˘ “ N p N ´ q µ s , we obtain E | g p X , X q| s “ ˆ N ˙ ´ ÿ ď k ă l ď N | g p x k , x l q| s ď s n s p n ´ q s ˆ NN ´ ˙ s ˆ N ˙ ´ ÿ ď k ă l ď N ` p x k ´ b q ` p x l ´ b q ` µ ˘ s ď s s ´ n s p n ´ q s ˆ NN ´ ˙ s ˆ N ˙ ´ ÿ ď k ă l ď N ` s ´ ` p x k ´ b q s ` p x l ´ b q s ˘ ` µ s ˘ ď s s ´ p s ` q n s p n ´ q s ˆ NN ´ ˙ s µ s . (3.19)Then we get from (3.18), (3.19) and (3.7) that b s ď s µ s ` µ ´ µ ˘ s { and g s ď s s ´ p s ` q ´ ´ nN ¯ s µ s ` µ ´ µ ˘ s { . The proof is completed. 11
Estimation of parameters
The jackknife variance estimator, defined by (3.10), is universal for U - and other statistics but it is notthe best for the particular ones. In Pumputis and ˇCiginas (2013), bootstrap and calibrated estimators,constructed for general L -statistics, are comparatively complex. Here, for both statistics of interest, wehave explicit expressions of their variances. Therefore, more natural as well as simple estimators of thevariances are possible. We give here, in fact, plug-in estimators of the variances, replacing populationmoments by their empirical counterparts in the parameters s and s defining variance (3.2). GMD statistic.
Denote D i : n “ X i ` n ´ X i : n and A i “ p i ´ n q{ n , for 1 ď i ď n ´
1. Then the estimators ofthe variance components (3.3) and (3.4) areˆ s G “ n ˆ NN ´ ˙ « n ´ ÿ i “ i p n ´ i q A i D i : n ` ÿ ď i ă j ď n ´ i p n ´ j q A i A j D i : n D j : n ff (4.1)and ˆ s G “ n p n ´ q NN ´ « n ´ ÿ i “ i p i ´ qp n ´ i ´ qp n ´ i q D i : n ` ÿ ď i ă j ď n ´ i p i ´ qp n ´ j ´ qp n ´ j q D i : n D j : n ff . (4.2)Denote by ˆ s G the estimator of the variance of U G obtained by plugging (4.1) and (4.2) into (3.2). Sample variance.
Denote the sample moments by m k “ n ´ ř ni “ p X i ´ n ´ ř nj “ X j q k , for k “ , . . . , s V “ n ˆ NN ´ ˙ ` m ´ m ˘ (4.3)and ˆ s V “ n p n ´ q N p N ´ qp N ´ q ˆ N ´ N ` N ´ m ´ m ˙ . (4.4)Let ˆ s V denote the estimator of the variance of U V obtained by plugging (4.3) and (4.4) into (3.2). In order to apply the one-term Edgeworth approximation (3.12) to the distribution functions of the statis-tics, the parameters a and k must be evaluated. Firstly, case (A), analogously to the variance estimationcase, we construct estimators of the parameters directly from the explicit expressions available. Secondly,case (B), we assume that the auxiliary variable z is at our disposal with the known values t z , . . . , z N u forall population elements. It is expected in this case, that z is well-correlated with the study variable x .Then the estimators below are immediately obtained from the true values of the parameters.12 MD statistic.
Case (A). With the notations used for the variance estimator, by formulas (3.13) and(3.14), the estimators areˆ a G “ ´ ˆ s ´ G n ˆ NN ´ ˙ « n ´ ÿ i “ i p n ´ i qp n ´ i q A i D i : n ` ÿ ď i ă j ď n ´ i p n ´ i qp n ´ j q A i A j D i : n D j : n ` ÿ ď i ă j ď n ´ i p n ´ j qp n ´ j q A i A j D i : n D j : n ` ÿ ď i ă j ă m ď n ´ i p n ´ j qp n ´ m q A i A j A m D i : n D j : n D m : n ff (4.5)and ˆ k G “ ´ ˆ s ´ G t n p n ´ q ˆ NN ´ ˙ n ´ ÿ i “ n ´ ÿ j “ n ´ ÿ m “ C i jm A j A m D i : n D j : n D m : n , (4.6)with the case function C i jm “ $’’’’’’’’’&’’’’’’’’’% i p i ´ qp n ´ m qr n ´ j ´ ` n ´ j p m ´ j qs , if i ď j ď m , i p i ´ qp n ´ j qr n ´ m ´ ` n ´ m p m ´ j qs , if i ď m ă j , j p n ´ m qrp i ´ qp n ´ i ´ q ` n ´ tp n ´ i qp n ´ i ´ qp i ´ j q ` i p i ´ qp m ´ i qus , if j ă i ă m , m p n ´ j qrp i ´ qp n ´ i ´ q ` n ´ t i p i ´ qp i ´ j q ` p n ´ i ´ qp n ´ i qp m ´ i qus , if m ă i ă j , j p n ´ i ´ qp n ´ i qr m ´ ` n ´ p n ´ m qp m ´ j qs , if j ă m ď i , m p n ´ i ´ qp n ´ i qr j ´ ` n ´ p n ´ j qp m ´ j qs , if m ď j ď i .Case (B). Having the additional information, the ordered sequence of the values z , . . . , z N is used insteadof x ď ¨ ¨ ¨ ď x N in the expressions (3.13) and (3.14) of the true parameters a and k . Denote the resultingestimates by z ˆ a G and z ˆ k G . Sample variance.
Case (A). From population parameters (3.15) and (3.16), we have the following plug-in estimators: ˆ a V “ ˆ s ´ V n ˆ NN ´ ˙ ` m ´ m m ` m ˘ (4.7)and ˆ k V “ ˆ s ´ V t n p n ´ q ˆ NN ´ ˙ N ´ ˆ ´p N ´ q m ´ N ´ N ´ m m ` NN ´ m ` m ˙ . (4.8)Case (B). In (3.15) and (3.16), the central population moments µ k are evaluated using the values z , . . . , z N .Then denote the new estimates by z ˆ a V and z ˆ k V . Replacing the population parameters a and k in Edgeworth expansion (3.12) by their estimators, weobtain the so-called empirical Edgeworth expansion. If the particular estimators of the parameters are13symptotically consistent, then, under the conditions of Theorem 2, the empirical Edgeworth expansionapproximates distribution function (3.9) with an error of the same order but in probability. In Bloznelis(2001), consistent jackknife estimators of the parameters were constructed. Bootstrap and calibrated esti-mators of the parameters were considered in ˇCiginas (2013a), and Pumputis and ˇCiginas (2013), respec-tively. Here, for each of the statistics U G and U V , we have two new versions of the empirical Edgeworthexpansion. GMD statistic.
By the results in Section 4.2, we have the empirical Edgeworth expansion p H nSG p y q “ F p y q ` ` ´ n { N ` p ´ n { N q y ˘ ˆ a G ` ` y ` ˘ ˆ k G t j p y q , (5.1)and, in the case where the auxiliary information is available, the approximation is z p H nSG p y q “ F p y q ` ` ´ n { N ` p ´ n { N q y ˘ z ˆ a G ` ` y ` ˘ z ˆ k G t j p y q , (5.2)which is not a random function because the values of the variable z are treated as fixed in the population. Sample variance.
The corresponding approximations to the distribution function of the Studentizedsample variance are p H nSV p y q “ F p y q ` ` ´ n { N ` p ´ n { N q y ˘ ˆ a V ` ` y ` ˘ ˆ k V t j p y q , (5.3)and z p H nSV p y q “ F p y q ` ` ´ n { N ` p ´ n { N q y ˘ z ˆ a V ` ` y ` ˘ z ˆ k V t j p y q , (5.4)where the later does not depend on the sample.Estimators of the parameters a and k in expansion (5.3) are asymptotically consistent under conditionsof Theorem 2. Efficiency of the other empirical Edgeworth expansions is examined in the simulationstudy in Section 6.It is known that, in general, non-parametric bootstrap approximations to distributions of statistics areusually of a similar accuracy as one-term Edgeworth expansions. We consider here the finite-populationbootstrap scheme introduced in Booth et al. (1994). We apply the results of Bloznelis (2007) where theaccuracy of this bootstrap method is considered for U -statistics.The bootstrap approximation to distribution (3.9) is constructed as follows. Write N “ kn ` l , where0 ď l ă n . Then, given the sample X , the empirical population r X of size N is formed by taking k copiesof X and, if l ą
0, adding the remaining l values which are the simple random sample Y “ t Y , . . . , Y l u drawn without replacement from the set X . With this particular bootstrap population r X , one can turnalready to an estimator of (3.9), despite that it is only the one of ` nl ˘ empirical populations. Next, wedraw the simple random sample r X “ t r X , . . . , r X n u without replacement from r X . Denote by r U “ U n p r X q the bootstrap estimator for the statistic of interest, and introduce the corresponding jackknife estimator r S “ S p r X q of the variance of r U under given population r X . Then the bootstrap approximation to (3.9) is r F nS p y q “ P t r U ´ E p r U | X , Y q ď y r S | X u , (5.5)which averages over all possible empirical populations. The following theorem is on the validity of thisapproximation for the statistics U G and U V . 14 heorem 3. Assume that the conditions of Theorem 2 are satisfied. Then, we have sup y P R | F nS p y q ´ r F nS p y q| “ o P p n ´ { ˚ q as n ˚ Ñ 8 , for U G and U V , respectively.Proof. It follows from condition (8) in Bloznelis (2007), that it suffices to verify that, for the statistics U G and U V , the moments E p X ´ X q and E p X ´ X q are bounded for all n ˚ , respectively. By theconditions of theorem, this requirement holds.Denote by r F nSG p y q and r F nSV p y q the bootstrap approximations for the statistics U G and U V , respectively. In this section, we illustrate the theoretical results on the second-order approximations to distributionfunctions of the Studentized GMD statistic and the Studentized sample variance by numerical exam-ples, according to the data framework in Section 2.2. Thus we consider also how outliers affect theseapproximations.For the statistics U G and U V , denote their ‘exact’ distribution functions by F nSG p y q and F nSV p y q , re-spectively. In the simulation experiments, these functions were evaluated by the Monte–Carlo method,drawing independently 10 samples without replacement from the population and using all values X , aswell as their bootstrap approximations based on the one (because of N “ kn ) empirical population r X constructed from the particular sample X . Denote true Edgeworth approximations (3.12) of the statisticsby H nSG p y q and H nSV p y q , respectively. To measure an efficiency of the empirical Edgeworth approxima-tions p H nSG p y q and p H nSV p y q , and the bootstrap approximations r F nSG p y q and r F nSV p y q , 10 samples withoutreplacement were drawn independently from the population.More specifically, in the tables below, the ‘exact’ distribution functions of the statistics, their normalapproximation, the true one-term Edgeworth expansions, the corresponding estimated Edgeworth approx-imations of two types, and the bootstrap approximations are represented by the several commonly used q -quantiles, q “ . , . , . , . , . , .
99. For the approximations, with the quantiles dependenton the sample, we give two characteristics of the efficiency: the empirical expectations p E p¨q and standarderrors p S p¨q from the realizations of these quantiles.Tables 3–6 present results of the approximations, where there are no outliers (the case of p { N “
0) inthe same underlying populations generated from the normal and gamma distribution in Section 2.2. Thecorrelation is r zx “ . H nSG p y q improves substantially the normal approxi-mation to F nSG p y q . With the help of the auxiliary information, H nSG p y q is estimated well by z p H nSG p y q .The bias of this estimate is small in comparison to a possible error of the estimator p H nSG p y q . But the laterimproves the normal approximation to the distribution of U G too. Differently from all other, the boot-strap approximation r F nSG p y q is almost unbiased, but its empirical quantiles have larger standard errorscompared to the empirical Edgeworth approximation. In Table 4, tendencies of the approximations tothe distribution function of U V are the same. In Tables 5–6, for the population from the gamma distribu-tion, the results are analogous to those in Tables 3–4, but all the corresponding approximations are lessaccurate. This is because of an asymmetry of the gamma distribution.15able 3: Approximations to F nSG p y q under N p , q with 0% outliers from N p , q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSG p q q « ´ . ´ . ´ .
363 1 .
223 1 .
546 2 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSG p q q « ´ . ´ . ´ .
357 1 .
214 1 .
536 2 . z p H ´ nSG p q q « ´ . ´ . ´ .
350 1 .
220 1 .
545 2 . E p H ´ nSG p q q « ´ . ´ . ´ .
353 1 .
217 1 .
541 2 . S p H ´ nSG p q q « .
034 0 .
023 0 .
016 0 .
013 0 .
021 0 . E r F ´ nSG p q q « ´ . ´ . ´ .
360 1 .
222 1 .
555 2 . S r F ´ nSG p q q « .
075 0 .
038 0 .
027 0 .
020 0 .
026 0 . Table 4: Approximations to F nSV p y q under N p , q with 0% outliers from N p , q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSV p q q « ´ . ´ . ´ .
477 1 .
160 1 .
461 2 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSV p q q « ´ . ´ . ´ .
447 1 .
145 1 .
432 1 . z p H ´ nSV p q q « ´ . ´ . ´ .
433 1 .
155 1 .
447 1 . E p H ´ nSV p q q « ´ . ´ . ´ .
431 1 .
157 1 .
450 1 . S p H ´ nSV p q q « .
046 0 .
038 0 .
029 0 .
020 0 .
032 0 . E r F ´ nSV p q q « ´ . ´ . ´ .
460 1 .
166 1 .
473 2 . S r F ´ nSV p q q « .
147 0 .
074 0 .
046 0 .
023 0 .
031 0 . Table 5: Approximations to F nSG p y q under G p , {? q with 0% outliers from G p , ? q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSG p q q « ´ . ´ . ´ .
443 1 .
188 1 .
503 2 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSG p q q « ´ . ´ . ´ .
413 1 .
172 1 .
468 1 . z p H ´ nSG p q q « ´ . ´ . ´ .
378 1 .
198 1 .
510 2 . E p H ´ nSG p q q « ´ . ´ . ´ .
407 1 .
177 1 .
476 1 . S p H ´ nSG p q q « .
055 0 .
045 0 .
033 0 .
024 0 .
037 0 . E r F ´ nSG p q q « ´ . ´ . ´ .
436 1 .
190 1 .
506 2 . S r F ´ nSG p q q « .
156 0 .
077 0 .
048 0 .
025 0 .
035 0 . Table 6: Approximations to F nSV p y q under G p , {? q with 0% outliers from G p , ? q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSV p q q « ´ . ´ . ´ .
699 1 .
109 1 .
391 1 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSV p q q « ´ . ´ . ´ .
561 1 .
077 1 .
324 1 . z p H ´ nSV p q q « ´ . ´ . ´ .
533 1 .
091 1 .
347 1 . E p H ´ nSV p q q « ´ . ´ . ´ .
529 1 .
097 1 .
355 1 . S p H ´ nSV p q q « .
075 0 .
077 0 .
067 0 .
040 0 .
060 0 . E r F ´ nSV p q q « ´ . ´ . ´ .
655 1 .
120 1 .
406 1 . S r F ´ nSV p q q « .
487 0 .
277 0 .
166 0 .
033 0 .
047 0 . F nSG p y q under N p , q with 6% outliers from N p , q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSG p q q « ´ . ´ . ´ .
553 1 .
143 1 .
434 1 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSG p q q « ´ . ´ . ´ .
490 1 .
118 1 .
388 1 . z p H ´ nSG p q q « ´ . ´ . ´ .
395 1 .
184 1 .
489 1 . E p H ´ nSG p q q « ´ . ´ . ´ .
470 1 .
132 1 .
410 1 . S p H ´ nSG p q q « .
069 0 .
063 0 .
050 0 .
033 0 .
051 0 . E r F ´ nSG p q q « ´ . ´ . ´ .
524 1 .
152 1 .
449 1 . S r F ´ nSG p q q « .
267 0 .
139 0 .
087 0 .
030 0 .
043 0 . Table 8: Approximations to F nSV p y q under N p , q with 6% outliers from N p , q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSV p q q « ´ . ´ . ´ .
100 1 .
037 1 .
280 1 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSV p q q « ´ . ´ . ´ .
740 0 .
979 1 .
181 1 . z p H ´ nSV p q q « ´ . ´ . ´ .
596 1 .
054 1 .
291 1 . E p H ´ nSV p q q « ´ . ´ . ´ .
634 1 .
036 1 .
266 1 . S p H ´ nSV p q q « .
086 0 .
098 0 .
095 0 .
050 0 .
075 0 . E r F ´ nSV p q q « ´ . ´ . ´ .
974 1 .
069 1 .
333 1 . S r F ´ nSV p q q « .
064 0 .
663 0 .
430 0 .
042 0 .
057 0 . Table 9: Approximations to F nSG p y q under G p , {? q with 6% outliers from G p , ? q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSG p q q « ´ . ´ . ´ .
546 1 .
148 1 .
445 1 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSG p q q « ´ . ´ . ´ .
483 1 .
124 1 .
395 1 . z p H ´ nSG p q q « ´ . ´ . ´ .
394 1 .
186 1 .
491 1 . E p H ´ nSG p q q « ´ . ´ . ´ .
470 1 .
134 1 .
410 1 . S p H ´ nSG p q q « .
064 0 .
060 0 .
049 0 .
032 0 .
048 0 . E r F ´ nSG p q q « ´ . ´ . ´ .
531 1 .
154 1 .
453 1 . S r F ´ nSG p q q « .
268 0 .
133 0 .
083 0 .
029 0 .
043 0 . Table 10: Approximations to F nSV p y q under G p , {? q with 6% outliers from G p , ? q , and r zx “ . q “ .
01 0 .
05 0 .
10 0 .
90 0 .
95 0 . F ´ nSV p q q « ´ . ´ . ´ .
042 1 .
045 1 .
296 1 . F ´ p q q « ´ . ´ . ´ .
282 1 .
282 1 .
645 2 . H ´ nSV p q q « ´ . ´ . ´ .
728 0 .
988 1 .
193 1 . z p H ´ nSV p q q « ´ . ´ . ´ .
597 1 .
055 1 .
292 1 . E p H ´ nSV p q q « ´ . ´ . ´ .
622 1 .
045 1 .
277 1 . S p H ´ nSV p q q « .
083 0 .
095 0 .
092 0 .
050 0 .
073 0 . E r F ´ nSV p q q « ´ . ´ . ´ .
944 1 .
079 1 .
347 1 . S r F ´ nSV p q q « .
184 0 .
651 0 .
416 0 .
041 0 .
058 0 . p { N “ .
06. In this case of outliers, the correspondingto Tables 3–6 results are given in Tables 7–10. A behaviour of the approximations to the distributions isvery similar to that in the case of no outliers, but erorrs of the approximations are larger now. One canobserve also that the estimates of the true Edgeworth expansions, which use the auxiliary information, aremuch more biased. It holds for the alternative empirical Edgeworth approximations too but in the caseof the statistic U V only (Tables 8 and 10). A sensitivity to the outliers is the smallest comparing Table 9with Table 5. The specific estimation strategies for scales are considered under simple random samples without re-placement. In a sense, they are consistent with the scale estimation by the sample variance. In particular,the proposed strategies ( S ) and ( S ) combine the use of the GMD statistic and its bias correction. Thiscombination allows an improvement of the scale estimation in populations where the part of outliers isnot large. As the numerical modeling indicates too, under ideal for the sample variance conditions (whenthere are no outliers), the efficiency of the strategies is not worse. It is important robustness property.The new estimators of the parameters and also empirical Edgeworth expansions for the GMD statisticand the sample variance are proposed using the detailed decompositions of the statistics. In general,well-correlated auxiliary information leads to effective inferences about the statistics of interest. References
Bickel, P.J., G ¨otze, F., van Zwet, W.R., 1986. The Edgeworth expansion for U -statistics of degree two. The Annalsof Statistics 14, 1463–1484.Bloznelis, M., 2001. Empirical Edgeworth expansion for finite population statistics I. Lithuanian MathematicalJournal 41, 120–134.Bloznelis, M., 2003. An Edgeworth expansion for Studentized finite population statistics. Acta Applicandae Math-ematicae 78, 51–60.Bloznelis, M., 2007. Bootstrap approximation to distributions of finite population U -statistics. Acta ApplicandaeMathematicae 96, 71–86.Bloznelis, M., G ¨otze, F., 1999. One-term Edgeworth expansion for finite population U -statistics of degree two.Acta Applicandae Mathematicae 58, 75–90.Bloznelis, M., G ¨otze, F., 2001. Orthogonal decomposition of finite population statistics and its applications todistributional asymptotics. The Annals of Statistics 29, 899–917.Booth, J.G., Butler, R.W., Hall, P., 1994. Bootstrap methods for finite populations. Journal of the American Statis-tical Association 89, 1282–1289.Chambers, R.L., 1986. Outlier robust finite population estimation. Journal of the American Statistical Association81, 1063–1069. hhikara, R.S., Feiveson, A.L., 1980. Extended critical values of extreme studentized deviate test statistics fordetecting multiple outliers. Communications in Statistics - Simulation and Computation 9, 155–166.ˇCiginas, A., 2012. An Edgeworth expansion for finite-population L -statistics. Lithuanian Mathematical Journal 52,40–52.ˇCiginas, A., 2013a. Second-order approximations of finite population L -statistics. Statistics 47, 954–965.ˇCiginas, A., 2013b. On the asymptotic normality of finite population L -statistics. Statistical Papers, pp. 1–12,doi:10.1007/s00362-013-0553-7.Erd˝os, P., R´enyi, A., 1959. On the central limit theorem for samples from a finite population. Publications of theMathematical Institute of the Hungarian Academy of Sciences 4, 49–61.Gini, C., 1912. Variabilit`a e mutabilit`a: contributo allo studio delle distribuzioni e delle relazioni statistiche. Cup-pini, Bologna.Glasser, G.J., 1962. Variance formulas for the mean difference and coefficient of concentration. Journal of theAmerican Statistical Association 57, 648–654.H´ajek, J., 1960. Limiting distributions in simple random sampling from a finite population. Publications of theMathematical Institute of the Hungarian Academy of Sciences 5, 361–374.Helmers, R., 1991. On the Edgeworth expansion and the bootstrap approximation for a Studentized U -statistic. TheAnnals of Statistics 19, 470–484.Hoeffding, W., 1948. A class of statistics with asymptotically normal distribution. The Annals of MathematicalStatistics 19, 293–325.Huber, P.J., 1981. Robust Statistics. Wiley, New York.Irwin, J.O., Kendall, M.G., 1944. Sampling moments of moments for a finite population. Annals of Eugenics 12,138–142.Kokic, P.N., Weber, N.C., 1990. An Edgeworth expansion for U -statistics based on samples from finite populations.Annals of Probability 18, 390–404.Lomnicki, Z.A., 1952. The standard error of Gini’s mean difference. The Annals of Mathematical Statistics 23,635–637.Nair, U.S., 1936. The standard error of Gini’s mean difference. Biometrika 28, 428–436.Pumputis, D., ˇCiginas, A., 2013. Estimation of parameters of finite population L -statistics. Nonlinear Analysis:Modelling and Control 18, 327–343.Putter, H., van Zwet, W.R., 1998. Empirical Edgeworth expansions for symmetric statistics. The Annals of Statistics26, 1540–1569.Serfling, R.J., 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York.Yitzhaki, S., Schechtman, E., 2013. The Gini Methodology: A Primer on a Statistical Methodology. Springer, NewYork. hao, L.C., Chen, X.R., 1990. Normal approximation for finite-population U -statistics. Acta Mathematicae Appli-catae Sinica 6, 263–272.-statistics. Acta Mathematicae Appli-catae Sinica 6, 263–272.