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).     

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.     

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.     

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.     

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.     

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.     

Sharp bounds on the expectations of second record values from symmetric populations

Combining the greatest convex minorant approximation (Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist., 24, 107–113.) with the Hölder inequality, we establish sharp bounds on the expectations of the second record statistics from symmetric populations. We also determine the distributions for which the bounds are attained. The optimal bounds are numerically evaluated and compared with other classical rough ones.     

Limit theory of bivariate dual generalized order statistics with random index

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.     

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.     

Upper non positive bounds on expectations of generalized order statistics from DD and DDA populations

We present the upper non positive bounds on the expectations of gOSs centered about the sample mean, which are based on the parent distributions with decreasing density and decreasing density on average distributions. Such bounds can be obtained only for particular cases of gOSs and they are expressed in units generated by the central absolute moments of a fixed order. The attainability conditions are also described. The method of deriving presented bounds is based on the maximization of appropriate norms over properly chosen convex sets. The paper complements the results of Bieniek [J. Statist. Plann. Inference, 2008; 138:971–981].     

On moments of dual generalized order statistics from Topp-Leone distribution

In this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.     

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

     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

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.     

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.     

Outliers and order statistics

Outliers are to be found among the extremes of a data set. Extremes are examples of order statistics. It is thus relevant to ask to what extent the statistical methods (and probabilistic properties) of outliers and of order statistics coincide and depend on each other. Whilst clear overlap is identifiable, aims and procedures are often quite distinct and each topic plays its own important role in the panoply of statistical principles and methodology.     

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.     

Dependence structure of generalized order statistics

In this article, simple expressions for marginal density functions of multiply censored generalized order statistics based on continuous distribution functions are obtained. Moreover, it is shown that generalized order statistics are multivariate totally positive and, thus, associated. This property is applied to show that regressions of generalized order statistics are nondecreasing under weak conditions.     

Conditional confidence interval estimation for the inverse weibull distribution based on censored generalized order statistics

This paper is concerned with the problem of obtaining the conditional confidence intervals for the parameters and reliability of the inverse Weibull distribution based on censored generalized order statistics, which are more general than the existing results in the literature. The coverage rate and the mean length of intervals have been obtained for different values of the shape parameter, via Monte Carlo simulation. Finally a numerical example is given to illustrate the inferential methods developed in this paper.