On distributions of order statistics and their applications to uniform distribution

Some results on the relationships between distributions of order statistics and of spacings are presented. These results are then used to establish a characterization of the uniform distribution extending some existing results in this direction.     

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.     

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 .     

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.     

Characterizations of the Pareto and related distributions

In this note, a class of Pareto distributions is characterized based on the Shannon entropy of k-record statistics. As a consequence of that characterizations of the uniform and exponential distributions are given. Received: October 1999     

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     

On approximating the Indirect Malmquist Productivity Indices by Fisher indices

In this paper it is shown that under certain assumptions the Cost Indirect Malmquist Productivity Index, which is a natural productivity measure for a cost constrained firm, can be approximated as the ratio of the Fisher output quantity index and the cost index.The views expressed in this paper are those of the author and do not necessarily reflect the policies of Statistics Netherlands.     

On the asymptotically uniform distribution modulo 1 of extreme order statistics

Let (X_m)∞₁ be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of rnax (X₁., X_n) to converge in distribution, as n ←∞ to a random variable with a uniform distribution on [0, 1).     

Some unified characterization results on the generalized Pareto distributions based on generalized order statistics

In the present study we extend and unify some existing results in the literature on characterization of the generalized Pareto distributions based on generalized order statistics.     

On the confidence region for the one-sample 156-1156-1156-1

Summary This article shows how the usual confidence region for the parameters of the multinomial distribution can be covered and closely approximated by an easily constructed polyhedron in the space of probability vectors.

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Zusammenfassung In dieser Arbeit wird gezeigt, wie der übliche Konfidenzbereich für die Parameter der Multinomialverteilung durch ein im Raum der Wahrscheinlichkeitsvektoren enthaltene leicht konstruierbare Polyeder überdeckt und gut approximiert werden kann.

     

On selecting the best ofk lognormal distributions

Summary Fork lognormal populations, which differ only in one certain parameter Ϙ, the problem of finding the population with the largest value ofϑ is considered. For two-parameter lognormal families, several natural choices ofϑ are treated, where the problem can be solved, through logarithmic transformation of the observations, within the framework of estimating parameters ink, possibly restricted, normal populations. For three-parameter lognormal families, this standard approach of selecting in terms of natural estimators fails to work ifϑ is the “guaranteed lifetime”. For this case, a selection procedure is derived which is based on anL-statistic which has the smallest asymptotic variance. Of importance here is that it is location equivariant, whereas it does not matter what it actually estimates. Comparisons are made with other suitable selection rules, through the asymptotic relative efficiencies, as well as in an example of intermediate sample sizes. It is shown that only in the latter, the selection rule, which is based on the sample minima, compares favorably. The research of this author was supported by the Office of Naval Research Contract N00014-88-K-0170 and NSF Grant Number DMS-8606964 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The research of this author was supported by the Air Force Office of Scientific Research Grant 85-0347 at the University of Illinois at Chicago.     

On the asymptotic Fisher information in order statistics

Metrika - We extend the result of Efron and Johnstone (1990), who expressed the Fisher information in terms of the hazard function, to express the Fisher information in order statistics as an...     

On bivariate generalized binomial and negative binomial distributions

The present paper deals with two types of generalized general binomial (binomial or negative binomial) distributions: (i) a univariate general binomial generalized by a bivariate distribution and (ii) a bivariate general binomial generalized by two independent univariate distributions. The probabilities, moments, conditional distributions and regression functions for these distributions are obtained in terms of bipartitional polynomials. Moreover recurrence relations for the probabilities and moments, independent of the bipartitional polynomials, are given. Finally these general results are applied to the (i) Binomial-Bivariate Poisson and (ii) Bivariate Binomial-Poissons distributions.     

On three and five parameter bivariate beta distributions

Summary In this paper two bivariate beta distributions have been studied. The five parameter bivariate beta distribution is derived from the Morgenstern-system of curves while the three parameter distribution is the bivariate Dirichlet distribution. In both cases the distributions of the product and the quotient of random variables are derived and other properties are also studied.     

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

Summary In 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.     

The geometry of log-normal and related distributions and an application to tracer-dilution curves

Summary
Most of a typical log-normal curve lies very close to two straight lines, namely the tangents at its points of inflection. This holds good for nearly all the rising part of the curve and for most of the top half beyond the maximum.
When log_e (t – to) is normally distributed with standard deviation , whether the whole curve can be observed or not, , to, the mean, the mode and the area under the whole curve can be derived from the triangle formed by these two tangents and the base line. The triangle also provides a simple test of goodness-of-fit. The corresponding inflection triangles of two other skew distributions are investigated in outline namely a chi² (or gamma) distribution and one arising in Brownian movement or random walk in one dimension with drift. These are compared with the log-normal one. The normal distribution and its inflection triangle are given as a limiting case of the log-normal one and it is shown that the curve on the cover of Statistica Neerlandica cannot be a normal one.
The application is to the extrapolation problem in heart dye-dilution curves, which record the distribution of passage times of an indicator through the lungs and heart. The usual semi-log extrapolation for determining the area under the complete curve is discussed critically; it is shown how the inflection triangle provides a simpler procedure that is probably more accurate. The general formulae, an outline of the application and numerical tables for 0 < 0.8 are given in the main text; the mathematical derivations of all the basic formulae are in an appendix (sections 11 to 16).
The information needed for practical applications without studying the theory is contained in sections 3 to 6 and the associated tables and figures.