On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

On characterizing distributions via regression of functions of generalized order statistics

Let X(1,n,m₁,k),X(2,n,m₂,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−∞≤a<b≤∞, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a⁺),g(b⁻) and E(g(X)). Then for some positive integer s,1<s≤n, we give characterization of distributions by means of

     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Domains of attraction of asymptotic distributions of extreme generalized order statistics

Abstract

In extreme value theory for ordinary order statistics, there are many results that characterize the domains of attraction of the three extreme value distributions. In this article, we consider a subclass of generalized order statistics for which also three types of limit distributions occur. We characterize the domains of attraction of these limit distributions by means of necessary and/or sufficient conditions for an underlying distribution function to belong to the respective domain of attraction. Moreover, we compare the domains of attraction of the limit distributions for extreme generalized order statistics with the domains of attraction of the extreme value distributions.     

Association of progressively Type-II censored order statistics

The association of progressively Type-II censored order statistics from a sample of associated random variables _X1,…,_Xn$X_1,\dots ,X_n$ is established. Moreover, some bivariate dependence properties are discussed for independent but not necessarily identically distributed _X1,…,_Xn$X_1,\dots ,X_n$.     

On generalized order statistics from linear exponential distribution and its characterization

In this paper, recurrence relations for single and product moments of generalized order statistics (gOSs) from linear exponential distribution (LE) are derived and characterizations of this distribution based on the conditional moments of the gOSs are given.     

Characterization of distributions by conditional expectation of generalized order statistics

In this work, general forms of many well-known continuous probability distributions are characterized by conditional expectation of some functions of generalized order statistics. These results are the generalization of the characterization results based on conditional expectation of the functions of order statistics given by Khan and Abu-Salih (1989).     

On generalized order statistics from generalized pareto distribution

The aim of this paper is to estimate parameters of generalized Pareto distribution based on generalized order statistics. Some non-Bayesian methods, such as MLE, bootstrap and unbiased estimators have been obtained to develop point and interval estimations. Bayesian estimations have also been derived under LSE and LINEX loss functions. To compare the performances of the employed methods, numerical results have been computed. To illustrate dependence and association properties of generalized order statistics, correlation coefficient and some informational measures in closed form have been obtained.     

Fisher information in generalized order statistics

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.     

Dependence Structure of Spacings of Order Statistics

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (X_k1:n, X_k2:n ? X_k1:n,…, X_kr:n ? X_kr?1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.     

Asymptotic random extremal ratio and product based on generalized order statistics and its dual

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.     

Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles

In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.     

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Fitting the generalized Pareto distribution to data based on transformations of order statistics

Generalized Pareto distribution (GPD) has been widely used to model exceedances over thresholds. In this article we propose a new method called weighted nonlinear least squares (WNLS) to estimate the parameters of the GPD. The WNLS estimators always exist and are simple to compute. Some asymptotic results of the proposed method are provided. The simulation results indicate that the proposed method performs well compared to existing methods in terms of mean squared error and bias. Its advantages are further illustrated through the analysis of two real data sets.     

Inference in a two-parameter generalized order statistics model

As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Utilizing this finding, a minimal sufficient and complete statistic is obtained along with distributional properties. Joint maximum likelihood estimation of the parameters is considered, and strong consistency and asymptotic efficiency of the estimator are established. A test is provided to decide whether a restriction to the submodel is reasonable.     

Inequalities for generalized order statistics from some restricted family of distributions

The Steffensen inequality is applied to derive quantile bounds for the expectations of generalized order statistics from a distribution belonging to a particular subclass of distributions. The subclass consists of F having the property that F^?1(0+)=x₀>0 and that x →[1? F(x)]x^z is nonincreasing for all x > X₀ and some z > 0.     

Projection bounds on expectations of generalized order statistics from DD and DDA families

We present sharp upper mean-variance bounds for expectations of generalized order statistics based on distributions coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with density decreasing on the average. The bounds are derived by application of the projection method.     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Generalized order statistics from arbitrary distributions and the Markov chain property

The joint and marginal distributions of generalized order statistics based on an arbitrary distribution function are established in terms of the lexicographic distribution function. Furthermore, we show that generalized order statistics and the corresponding number of ties form a two-dimensional Markov chain.