Some characterizations of families of distributions,including logistic and exponential ones,by properties of order statistics

New characterizations of distributions based on properties of the maximal order statistics are obtained. The families of distributions that are characterized by some properties of maxima include exponential and logistic distributions as partial cases. Bibliography: 4 titles.     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.     

Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X ₁, ...,X _n) (Y ₁ ...,Y _n) of nonnegative random vectors by requiring thek-th order statistic ofa ₁ X ₁,..., a _n X _n to be stochastically smaller than thek-th order statistic ofa ₁ Y ₁, ...,a _n Y _n for all choices ofa _i >0,i=1, 2, ...,n. We identify a class of functionsM _{k, n} such that if and only ifE(X)E(Y) for allM _k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.     

Characterizations of mixture of two-component exponentiated family of distributions based on generalized order statistics

In this paper, we shall characterize mixture of two components of exponentiated family of distributions based on recurrence relations for moment and conditional moment generating functions of generalized order statistics. Results for ordinary order statistics and upper kth record values are obtained as special cases.     

Characterization of distributions by equalities of order statistics

It is proved that equality of two different order statistics holds for degenerate or two-point distributions only. Several examples of such equalities are considered, and particular values of the corresponding distribution functions are obtained. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 168–173.     

Characterization of distributions by the expected values of the order statistics

Annals of the Institute of Statistical Mathematics -     

On a logarithmic integral and the moments of order statistics from the Weibull-geometric and half-logistic families of distributions

We provide an explicit analytical solution for a logarithmic integral in terms of the Lerch transcendent function together with the generalized Stirling numbers of the first kind. For some special cases of interest in statistical applications, the explicit solution can be expressed in terms of the polylogarithm function together with the aforementioned Stirling numbers. As a consequence, we obtain explicit expressions for the moments of order statistics from the half-logistic distribution, the Weibull-geometric distribution and the long-term Weibull-geometric distribution, which include as particular cases the extended exponential-geometric distribution and the long-term extended exponential-geometric distribution, among others. These analytical expressions are useful for computational purposes.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

Limiting properties of order statistics

We consider the relationship between the convergence of the distribution functions of sums of independent random variables and the asymptotic distribution of order statistics.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 128–144, 1986.     

Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter P. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, J. Statist. Comput. Simulation, 52, 385–398).Earlier went by the name R. A. Sandhu.     

Characterizations of distributions by moment properties of sums of records and successive maxima

We consider the estimation problem for expectations of random variables that are sums of records and successive maxima. The corresponding upper bounds and characterizations of source distributions are obtained in the case where these bounds are attained, Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov; POMI, Vol. 361, 2008, pp. 138–144.     

On a characterization of the exponential distribution by properties of order statistics

A continuous c.d.f. F(x), strictly increasing for x > 0, is exponential if and only if X_{j, n} - X_{i, n} and X_{j−i, n−i} have identical distribution for some i, n, j = j₁, j₂, 1 ⩽ i < j₁ <j₂ ⩽ n, n ⩾ 3. A new proof of this characterization is given, since in Ahsanullah (1975) where it was stated first, an implicit assumption in the proof is that F is NBU or NWU.     

Characterizations of natural exponential families with power variance functions by zero regression properties

Summary Series of new characterizations by zero regression properties are derived for the distributions in the class of natural exponential families with power variance functions. Such a class of distributions has been introduced in Bar-Lev and Enis (1986) in the context of an investigation of reproductible exponential families. This class is broad and includes the following families: normal, Poisson-type, gamma, all families generated by stable distributions with characteristic exponent an element of the unit interval (among these are the inverse Gaussian, Modified Bessel-type, and Whittaker-type distributions), and families of compound Poisson distributions generated by gamma variates. The characterizations by zero regression properties are obtained in a unified approach and are based on certain relations which hold among the cumulants of the distributions in this class. Some remarks are made indicating how the techniques used here can be extended to obtain characterizations of general exponential families.The work of this author was performed while he was a visitor in the Department of Statistics, State University of New York at Buffalo