Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

Sharp bounds on the expectations of L-statistics expressed in the Gini mean difference units

We describe a method of calculating sharp lower and upper bounds on the expectations of arbitrary, properly centered L-statistics expressed in the Gini mean difference units of the original i.i.d. observations. Precise values of bounds are derived for the single-order statistics, their differences, and some examples of L-estimators. We also present the families of discrete distributions which attain the bounds, possibly in the limit.     

Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis

Estimates based on L-moments are less non-robust than estimates based on ordinary moments because the former are linear combinations of order statistics for all orders, whereas the later take increasing powers of deviations from the mean as the order increases. Estimates based on L-moments can also be more efficient than maximum likelihood estimates. Similarly, L-skewness and L-kurtosis are less non-robust and more informative than the traditional measures of skewness and kurtosis. Here, we give nonparametric bias-reduced estimates of both types of skewness and kurtosis. Their asymptotic computational efficiency is infinitely better than that of corresponding bootstrapped estimates.     

Best linear unbiased interpolation of order statistics

We derive the best linear unbiased interpolation for the missing order statistics of a random sample using the well-known projection theorem. The proposed interpolation method only needs the first two moments on both sides of a missing order statistic. A simulation study is performed to compare the proposed method with a few interpolation methods for exponential and Lévy distributions.     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

Distributed inference for two-sample U-statistics in massive data analysis

This paper considers distributed inference for two-sample U-statistics under the massive data setting. In order to reduce the computational complexity, this paper proposes distributed two-sample U-statistics and blockwise linear two-sample U-statistics. The blockwise linear two-sample U-statistic, which requires less communication cost, is more computationally efficient especially when the data are stored in different locations. The asymptotic properties of both types of distributed two-sample U-statistics are established. In addition, this paper proposes bootstrap algorithms to approximate the distributions of distributed two-sample U-statistics and blockwise linear two-sample U-statistics for both nondegenerate and degenerate cases. The distributed weighted bootstrap for the distributed two-sample U-statistic is new in the literature. The proposed bootstrap procedures are computationally efficient and are suitable for distributed computing platforms with theoretical guarantees. Extensive numerical studies illustrate that the proposed distributed approaches are feasible and effective.     

The beta exponentiated Weibull distribution

The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of the order statistics can also be expressed as a linear combination of Weibull densities. We obtain the moments of the order statistics. The expected information matrix is derived. We define a log-beta exponentiated Weibull regression model to analyse censored data. The estimation of the parameters is approached by the method of maximum likelihood. The usefulness of the new distribution to analyse positive data is illustrated in two real data sets.     

On the exact distribution of the sum of the largest n−k out of n normal random variables with differing mean values

Motivated by an application in Electrical Engineering, we derive the exact distribution of the sum of the largest n?k out of n normally distributed random variables, with differing mean values. Comparisons are made with two normal approximations to this distribution—one arising from the asymptotic negligibility of the omitted order statistics and one from the theory of L-statistics. The latter approximation is found to be in excellent agreement with the exact distribution.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right-censored order statistics from a generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.     

Confidence intervals based on L-moments for quantiles of the GP and GEV distributions with application to market-opening asset prices data

In a ground-breaking paper published in 1990 by the Journal of the Royal Statistical Society, J.R.M. Hosking defined the L-moment of a random variable as an expectation of certain linear combinations of order statistics. L-moments are an alternative to conventional moments and recently they have been used often in inferential statistics. L-moments have several advantages over the conventional moments, including robustness to the the presence of outliers, which may lead to more accurate estimates in some cases as the characteristics of distributions. In this contribution, asymptotic theory and L-moments are used to derive confidence intervals of the population parameters and quantiles of the three-parametric generalized Pareto and extreme-value distributions. Computer simulations are performed to determine the performance of confidence intervals for the population quantiles based on L-moments and to compare them to those obtained by traditional estimation techniques. The results obtained show that they perform well in comparison to the moments and maximum likelihood methods when the interest is in higher quantiles, or even best. L-moments are especially recommended when the tail of the distribution is rather heavier and the sample size is small. The derived intervals are applied to real economic data, and specifically to market-opening asset prices.     

On the asymptotic normality of finite population L-statistics

We give sufficient conditions for the asymptotic normality of linear combinations of order statistics ( \(L\) -statistics) in the case of simple random samples without replacement. In the first case, restrictions are imposed on the weights of \(L\) -statistics. The second case is on trimmed means, where we introduce a new finite population smoothness condition.     

Asymptotic expansions of moments and cumulants

This paper shows how procedures for computing moments and cumulants may themselves be computed from a few elementary identities.Many parameters, such as variance, may be expressed or approximated as linear combinations of products of expectations. The estimates of such parameters may be expressed as the same linear combinations of products of averages. The moments and cumulants of such estimates may be computed in a straightforward way if the terms of the estimates, moments and cumulants are represented as lists and the expectation operation defined as a transformation of lists. Vector space considerations lead to a unique representation of terms and hence to a simplification of results. Basic identities relating variables and their expectations induce transformations of lists, which transformations may be computed from the identities. In this way procedures for complex calculations are computed from basic identities.The procedures permit the calculation of results which would otherwise involve complementary set partitions, k-statistics, and pattern functions. The examples include the calculation of unbiased estimates of cumulants, of cumulants of these, and of moments of bootstrap estimates.     

Second-order and resampling approximation of finite population U-statistics based on stratified samples

We construct one-term Edgeworth expansions to distributions of U statistics and Studentized U-statistics, based on stratified samples drawn without replacement. Replacing the cumulants defining the expansions by consistent jackknife estimators, we obtain empirical Edgeworth expansions. The expansions provide second-order approximations that improve upon the normal approximation. Theoretical results are illustrated by a simulation study where we compare various approximations to the distribution of the commonly used Gini's mean difference estimator.     

The beta generalized Rayleigh distribution with applications to lifetime data

For the first time, we propose a new distribution so-called the beta generalized Rayleigh distribution that contains as special sub-models some well-known distributions. Expansions for the cumulative distribution and density functions are derived. We obtain explicit expressions for the moments, moment generating function, mean deviations, Bonferroni and Lorenz curves and densities of the order statistics and their moments. We estimate the parameters by maximum likelihood and provide the observed information matrix. The usefulness of the new distribution is illustrated through two real data sets that show that it is quite flexible in analyzing positive data instead of the generalized Rayleigh and Rayleigh distributions.     

Testing for Bivariate Extreme Dependence Using Kendall's Process

Abstract. New tests for the hypothesis of bivariate extreme‐value dependence are proposed. All test statistics that are investigated are continuous functionals of either Kendall's process or its version with estimated parameters. The procedures considered are based on linear combinations of moments and on Cramér–von Mises distances. A suitably adapted version of the multiplier central limit theorem for Kendall's process enables the computation of asymptotically valid p‐values. The power of the tests is evaluated for small, moderate and large sample sizes, as well as asymptotically, under local alternatives. An illustration with a real data set is presented.     

Bounds for L-Statistics from Weakly Dependent Samples of Random Length

ABSTRACT

Sharp bounds on expected values of L-statistics based on a sample of possibly dependent, identically distributed random variables are given in the case when the sample size is a random variable with values in the set {0, 1, 2,…}. The dependence among observations is modeled by copulas and mixing. The bounds are attainable and provide characterizations of some non trivial distributions.     

Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics

We find the distribution that has maximum entropy conditional on having specified values of its first r  L$L$-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics.     

Some Results for Beta Fréchet Distribution

Nadarajah and Gupta (2004 Nadarajah , S. , Gupta , A. K. ( 2004 ). The beta Fréchet distribution . Far East J. Theoret. Statist. 14 : 15 – 24 . [Google Scholar]) introduced the beta Fréchet (BF) distribution, which is a generalization of the exponentiated Fréchet (EF) and Fréchet distributions, and obtained the probability density and cumulative distribution functions. However, they did not investigate the moments and the order statistics. In this article, the BF density function and the density function of the order statistics are expressed as linear combinations of Fréchet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Fréchet distribution. We derive explicit expansions for the ordinary moments and L-moments and obtain the order statistics and their moments. We also discuss maximum likelihood estimation and calculate the information matrix which was not given in the literature. The information matrix is numerically determined. The usefulness of the BF distribution is illustrated through two applications to real data sets.