A note on an approximation to neyman type a distributions

This note outlines a steepest descent approximation to the tail probability of the Neyman type A distribution.     

Goodness of fit tests for discrete distributions

Some alternative procedures for testing goodness of fit in discrete distributions are discussed here.. These procedures are based on the probability generating functions.. The methods considered are quite general, being applicable in multidimensional situations., The strength of the tests lies in that no ambiguity as to classification of the data arises.. Hov-ever, some difficulties in the proposed procedures are also pointed out.     

A note on gamma difference distributions

It is the aim of this note to point out that the double gamma difference distribution recently introduced by [Augustyniak M, and Doray, LG. Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variables. J Statist Comput Simul. 2012;82:1621–1634] is well known in financial econometrics: it is the symmetric variance gamma family of distributions. We trace back to the various origins of this distribution. In addition, we consider in some detail the difference of two independent gamma distributed random variables with different shape parameters.     

A note about calibrated prediction regions and distributions

In this work we address the problem of the construction of prediction regions and distribution functions, with particular regard to the multidimensional setting. Firstly, we define a simple procedure for calculating the predictive distribution function which gives improved prediction limits. Secondly, with a multivariate generalization of a result presented in Ueki and Fueda (2007), we propose a method for correcting estimative prediction regions, to reduce their coverage error to the third-order accuracy. The improved prediction regions and the associated distribution functions are easy to calculate using a suitable bootstrap procedure. Examples of application are included, showing the good performance of the proposed method, even if we consider an approximated model for prediction purposes.     

Discrete distributions based on inter arrival times with application to football data

Motivated by McShane, Adrian, Bradlow and Fader [Journal of Business and Economic Statistics, 26, 2008, 369–378], we introduce thirteen discrete distributions. We give explicit expressions for their probability mass functions. We analyze two football data sets and show that some of the proposed distributions provide better fits than the distribution due to McShane, Adrian, Bradlow, and Fader.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function     

A note on the Cauchy-type mixture distributions

An interesting class of continuous distributions, called Cauchy-type mixture, with potential applications in modelling erratic phenomena is introduced by Soltani and Tafakori [A class of continuous kernels and Cauchy type heavy tail distributions. Statist Probab Lett. 2013;83:1018–1027]. In this work, we provide more insights into the Cauchy-type mixture distributions, involving certain characterizations, connections with the generalized Linnik distributions and the class of discrete distributions induced by stable laws. We also prove that the Laplace transform of Cauchy-type mixture distributions when normalized by constant terms become as a density functions in terms of distributional conjugate property.     

A note on the joint distribution of correlated quadratic forms

In the present paper, the authors obtained exact and asymptotic expressions for the joint distribution of correlated quadratic forms when the underlying distribution is a multivariate normal.     

A note on an IPPS sampling scheme

Summary This paper presents a new IPPS sampling scheme possessing some desirable properties and providing an unbiased and non-negative variance estimator under H-T model. An empirical study is also undertaken to examine the performance of the scheme compared to some standard sampling schemes. The authors are grateful to the referees for providing many helpful suggestions on the earlier draft of the paper.     

Characterization of the quantitative-qualitative measure of inaccuracy for discrete generalized probability distributions

A characterization of the quantitative-qualitative measure of inaccuracy depending upon the additivity postulate and the mean value property for discrete generalized probability distribution has been provided.     

On k-distorted generalized discrete family of distributions

We propose a new generalized discrete family of distributions which permits inflation/deflation at any single point in the support of distribution. Using the same, a zero-distorted generalized discrete family of distributions is introduced and some of its properties are studied. As an illustration, we study in detail, zero-distorted generalized Poisson distribution. Real-life applications of this distribution using well-known data sets are reported, which include an actuarial application of the proposed model.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

A note on the inequalities for tail probability of the multivariate normal distribution

In this paper inequalities given by Harkness Godambe (1976) for the rail probabilities of the multivariate normal distribution in the equicorrelated case are improved by using the properties of the characteristic roots of a matrix and of the convex function.     

Characterizations of discrete compound Poisson distributions

ABSTRACT

The aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.     

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.     

The discrete delta and nabla Mittag-Leffler distributions

In this paper, we extend Bernstein theorem by using basic tools of calculus on time scales, and, as a further application of it, the discrete nabla and delta Mittag-Leffler distributions are introduced here with respect to their Laplace transforms on the discrete time scale. For these discrete distributions, infinite divisibility and geometric infinite divisibility are proved along with some statistical properties. The delta and nabla Mittag-Leffler processes are defined.     

A new goodness-of-fit test for certain bivariate distributions applicable to traffic accidents

T. Cacoullos and H. Papageorgiou [On some bivariate probability models applicable to traffic accidents and fatalities, Int. Stat. Rev. 48 (1980) 345–356] studied a special class of bivariate discrete distributions appropriate for modeling traffic accidents, and fatalities resulting therefrom. The corresponding random variable may be written as $Z={(N,Y)}^{\prime }$, with $Y={\sum }_{j=1}^N{X}_j$, where ${\left\{X_j\right\}}_{j=1}^N$, are independent copies of a (discrete) random variable $X$, and $N$ is independent of ${\left\{X_j\right\}}_{j=1}^N$, and follows a Poisson law. If $X$ follows a Poisson law (resp. Binomial law), the resulting distribution is termed Poisson–Poisson (resp. Poisson–Binomial). L2-type goodness-of-fit statistics are constructed for the ‘general distribution’ of this kind, where $X$ may be an arbitrary discrete nonnegative random variable. The test statistics utilize a simple characterization involving the corresponding probability generating function, and are shown to be consistent. The proposed procedures are shown to perform satisfactorily in simulated data, while their application to accident data leads to positive conclusions regarding the modeling ability of this class of bivariate distributions.