Udo Kamps

A concept of generalized order statistics

A form of the joint distribution of n ordered random variables is presented that enables a unified approach to a variety of models of ordered random variables, e.g. order statistics and record values. Several other models are shown. In particular, sequential order statistics are introduced as a modification of order statistics which is naturally suggested by a statistical application in reliability theory. In the distribution theoretical sense, all of these models of ordered random variables are contained in the proposed concept of generalized order statistics. Numerous related results on distributional and moment properties of ordinary order statistics and record values are found in the literature which are deduced separately. Generalized order statistics, however, provide a suitable approach to explain these similarities and analogies in the two models and to generalize related results. Through integration of known properties the structure of the embedded models becomes clearer. On the other hand, we obtain the validity of these properties and their generalizations for generalized order statistics, and hence for different models of ordered random variables. In the present paper we develop the distribution theory for generalized order statistics. Representations for the one-, two- and higher-dimensional marginal densities and a form of the one-dimensional marginal distribution functions are given as well as recurrence relations for marginal densities and distribution functions. Moreover, we give representations for moments and differences of moments of generalized order statistics, sufficient conditions for the existence of moments, and we show some explicit expressions for the moments of generalized order statistics based on power function, Pareto and Weibull distributions.

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.

Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates

Abstractk-out-of-n systems frequently appear in applications. They consist of n components of the same kind with independent and identically distributed life-lengths. The life-length of such a system is described by the (n−k+1)-th order statistic in a sample of size n when assuming that remaining components are not affected by failures. Sequential order statistics are introduced as a more flexible model to describe ‘sequential k-out-of-n systems’ in which the failure of any component possibly influences the other components such that their underlying failure rate is parametrically adjusted with respect to the number of preceding failures. Useful properties of the maximum likelihood estimators of the model parameters are shown, and several tests are proposed to decide whether the new model is the more appropriate one in a given situation. Moreover, for specific distributions, e.g. Weibull distributions, simultaneous maximum likelihood estimation of the model parameters and distribution parameters is considered.

Bounds for means and variances of progressive type II censored order statistics

By applying different methods, bounds for expected values and variances of progressive type II censored order statistics are derived. Since ordinary order statistics are contained in the model, well-known bounds for their moments are obtained as particular cases. The method of the greatest convex minorant leads to close bounds for means of progressive type II censored order statistics, which are even new in the particular set-up of ordinary order statistics. Numerical examples are shown in order to compare bounds and exact values for means w.r.t. underlying rectangular and normal distributions.

Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n−k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)

Relations for expectations of functions of generalized order statistics

Relations are derived for expectations of functions of generalized order statistics within a class of distributions including a variety of identities for single and product moments of ordinary order statistics and record values as particular cases. Since several models of ordered random variables are contained in the concept of generalized order statistics and since there are no restrictions imposed on their parameters, the identities can be applied to all of these models with their different interpretations.

Unimodality of uniform generalized order statistics, with applications to mean bounds

We prove that uniform generalized order statistics are unimodal for an arbitrary choice of model parameters. The result is applied to establish optimal lower and upper bounds on the expectations of generalized order statistics based on nonnegative samples in the population mean unit of measurement. The bounds are attained by two-point distributions.

A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations

SummaryIn a class of distribution functions, including exponential, power function, Pareto, Lomax, and logistic distributions, a general recurrence relation for moments of order statistics is given. The validity of this identity for certain constants and some sequence of order statistics leads to characterizations of probability distributions. Several recurrence relations and characterization results known from the literature are particular cases of the theorems stated.

On a class of premium principles including the Esscher principle

Abstract A class of premium calculation principles is considered with the premiums obtained as expected values of suitably transformed distribution functions. The Esscher principle is a particular example. It is found that the likelihood ratio ordering of risks is preserved for any of these principles. A renewal theoretic interpretation of a special principle is given, and useful properties as well as a related characterization of the exponential distribution are shown.