Some notes on the non-central negative binomial distribution

Summary In this paper we shall record some facts and further examine certain properties of the non-central negative binomial (NNB) distribution (Laguerre series distribution of Gurland, Chen and Hernandez 1983). We consider, among others, a stochastic formulation (birth-and-death process), the series expansion of the probability distribution and the corresponding series expansion of a generalized exponential distribution (Ong/Lee 1986), and the connection of the NNB distribution with the non-central beta, gamma and non-central gamma distributions. A four-parameter version of the NNB distribution is also presented.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.     

Saddlepoint approximations to the mean and variance of the extended hypergeometric distribution

Conditional inference on 2 x 2 tables with fixed margins and unequal probabilities is based on the extended hypergeometric distribution. If the support of the distribution is large, exact calculation of the conditional mean and variance of the table entry may be computationally demanding. This paper proposes a single‐saddlepoint approximation to the mean and variance. While the approximation achieves acceptable accuracy for ordinary practical purposes, an alternative saddlepoint approximation is provided that gives much closer to exact results. It improves the accuracy of current approximations to up to more than four powers of ten.     

Some properties of the minimum and the maximum of random variables with joint logconcave distributions

It is shown that if (X ₁, X ₂, . . . , X _n ) is a random vector with a logconcave (logconvex) joint reliability function, then X _P = min _i∈P X _i has increasing (decreasing) hazard rate. Analogously, it is shown that if (X ₁, X ₂, . . . , X _n ) has a logconcave (logconvex) joint distribution function, then X ^P  = max _i∈P X _i has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li (Metrika 65:325–330, 2007). It is also shown that if (X ₁, X ₂, . . . , X _n ) has an exchangeable logconcave density function then both X _P and X ^P have increasing likelihood ratio.     

Characterizations of negative multinomial distributions based on conditional distributions

In this paper characterizations of negative multinomial distributions based on conditional distributions have been studied.     

A note on a short table of the generalized hypergeometric distribution

In a paper byMathai/Saxena [1968, 21–39], the present authors found some entries with error and others with lack of precision. Recently,Conde/Kalla [1979] have tabulated₂ F ₁ (a, b; c; x) fora=0.5 (0.5) 5.0,b=0.5 (0.5) 5.0,c=0.5 (0.5) 12 andx=–2.50 (0.05) 0.95. The present note explains the method of computation and checks applied, Wronskians, continuation formulae, special cases etc. to insure the stated accuracy.     

Some properties of the distribution of an operational ridge estimator

Metrika - The present paper investigates the properties of the sampling distribution of an operational general ridge estimator. Exact expression are given for the moments and their approximations...     

Conditional distributions and the bivariate normal distribution

Summary An alaysis of the extent to which conditional distributions of a bivariate vector characterize bivariate normality is given.     

Three-stage estimation procedures for the negative exponential distributions

Fixed-width confidence interval estimation problems for location parameters of negative exponential populations have been studied. Three-stage sampling procedures have been developed for both the one- and two-sample situations. Our discussions are primarily concerned with second-order expansions of various characteristics of the proposed procedures including those for the achieved coverage probability in either problem. Some simulated results are also presented to indicate the usefulness of our procedures for moderate sample sizes.     

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.     

Bounds for the difference between median and mean of beta and negative binomial distributions

Summary We obtain bounds for the difference between the median and mean of the beta and negative binomial distributions using elementary methods.     

Sampling distributions associated with the multivariate t distribution

The known sampling distributions and simulation methods associated with multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.     

Bounds for the median of the negative binomial distribution

Summary Bounds are obtained for the median of the negative binomial distribution which are valid for all possible parameter values of the distribution when the median is defined as inf {x: P(X≤x)≥1/2}.     

The estimation of the two-dimensional distribution of a statistical set, based on the marginal distributions of its strata

In dit artikel wordt een methode ontwikkeld, met behulp waarvan men de simul-tane verdelingsfunctie F(x,y) van twee stochastische variabelen x en y kan aan-passen, voor het geval dat de steekproeven van x en y onafhankelijk van elkaar ge-nomen zijn.
De methode is gebaseerd op de veronderstelling dat de gehelepopulatie, waarop x en y betrekking hebben, bestaat uit een groot aantal subpopulaties; voor deze subpopulaties moet gelden dat de stochastische variabelen x en y dezelfde voor-waardelijke verdelingsdichtheid f( x/y ) hebben, terwijl alle voorwaardelijke ver-delingsdichtheden f( y / x ) verschillend moeten zijn. Daarnaast wordt aangegeven hoe men uit dezelfde gegevens de correlatiecoëfficient van x en y kan schatten. Als analogon van deze "enkelvoudige" correlatiecoefficient wordt de z.g. collec-tieve correlatiecoëfficient van twee stochastische vectoren gedefinieerd, terwijl tenslotte wordt gedemonstreerd hoe het een en ander kan worden toegepast bij de voorspelling van het toekomstige verloop van tijdreeksen.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Some characterizations of the bivariate normal distribution

Summary In this note assuming the conditional normality of one component of a bivariate random vector given the value of the other component some characterizations of the joint normality of the random vector are given.