On characterizing distributions by conditional expectations of functions of order statistics

In the present paper, we give some general theorems on characterizations based on conditional expectations of the functions of order statistics. In addition, we indicate special forms of the theorems for the familiar probability distributions.     

Sharp bounds on moments of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of rearranged Moriguti's inequality. The method yields improvements of known quantile and moment bounds for expectations of order and record statistics based on independent identically distributed random variables. The bounds are attainable providing new characterizations of two-point distributions. Received: January 1999     

Variation diminishing property of densities of uniform generalized order statistics

Let f _*,r , r ≥ 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0,1) of any linear combination does not exceed the number of sign changes in the sequence (a ₁, . . . ,a _r ). This result is applied to study monotonicity and convexity properties of f _*,r .     

Exponential inequalities for tail probabilities of generalized order statistics

We establish exponential bounds for the probability that a generalized order statistic exceeds a given threshold or falls below a given threshold. As a main tool we apply the Bernstein inequality for sums of independent random variables.Received May 2003     

On characterizations of distributions by expected values of order statistics and record values with gap

In this paper, we obtain some recurrence relationships for conditional expectations of nonadjacent order statistics and record values when the distribution function is absolutely continuous, and we prove that the distribution function is uniquely determined by the distribution of conditioned record values and by the expected values of these records. Further, different distributions are characterized by these relationships.     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

Sharp exponential and entropy bounds on expectations of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of Moriguti's inequality combined with the Young inequality. The bounds are expressed in terms of exponential moments or entropy. They are attainable providing new characterizations of some nontrivial distributions. Received October 2001/Revised May 2002     

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

Predicting the sample mean by extreme order statistics

Summary This paper considers the prediction of the sample mean by extreme order statistics when the population distribution is known. The predictor and its mean square error are found. The problem is studied in details for the normal model.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

Tail index estimation based on linear combinations of intermediate order statistics

Based on linear combinations of intermediate order statistics, we introduce a new class of estimators for the exponent of a distribution function F with a regularly varying upper tail. We prove asymptotic normality and we make a comparison with existing proposals using the mean squared error as criterion.     

Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous exponentiated gamma components. The results established here are developed in two directions. First, when a system possibly has different shape and scale parameters and the matrix of those different parameters following the chain majorization order, we study the reversed hazard rate order of parallel systems and the usual stochastic order of series systems. Next, by using the concept of vector majorization, we establish the usual stochastic order of series systems and the reversed hazard rate order of parallel systems.     

Estimation of the parameters of log-gamma distribution using order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of log-gamma distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Marginal distributions of sequential and generalized order statistics

Expressions for marginal distribution functions of sequential order statistics and generalized order statistics are presented without any restrictions imposed on the model parameters. The results are related to the relevation transform, to the distribution of the product of Beta distributed random variables, and to Meijers G-functions. Some selected applications in the areas of moments, conditional distributions, recurrence relations, and reliability properties are shown. Key words:Order statistics; Generalized order statistics; Sequential order statistics; Record values; Distribution theory; Meijers G-function; Recurrence relations; Reliability properties.     

Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998     

An identity for expectations of functions of order statistics

Summary We consider an identity for expectations of general functions of order statistics valid in a parametric class of probability distributions. Corresponding characterization results are indicated.     

Mean residual life functions of finite mixtures,order statistics and coherent systems

We study properties of the mean residual life functions of finite mixtures. Specifically, we study ordering properties, monotonicity and the limiting behaviour. We show, under some mild conditions, that the limiting behaviour is similar to that of the strongest member (in the mean residual life order) of the mixture. We also consider the case of negative mixtures (i.e., mixtures with some negative coefficients) which is applied to study the behaviour of the mean residual life of order statistics and coherent systems with possibly dependent components. Partially Supported by Ministerio de Ciencia y Tecnologí a under grant BFM2003-02947 and Fundación Séneca under grant 00698/PI/04.     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

Fisher information in record data

We present some general results on Fisher information (FI) contained in upper (or lower) record values and associated record times generated from a sequence of i.i.d. continuous variables. For the record data obtained from a random sample of fixed size, we establish an interesting relationship between its FI content and the FI in the data consisting of sequential maxima. We also consider the record data from an inverse sampling plan (Samaniego and Whitaker, 1986). We apply the general results to evaluate the FI in upper as well as lower records data from the exponential distribution for both sampling plans. Further, we discuss the implication of our results to statistical inference from these record data. Received: December 2001 Acknowledgements. This research was supported by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grants 7990089 and 1010222 of Chile. We would like to thank the Department of Statistics at the University of Concepción for its hospitality during the stay of H. N. Nagaraja in Chile in March of 2000, when the initial work was done. We are grateful to the two referees for various comments that let to improvements in the paper.