A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.     

The non-central negative binomial distribution: Further properties and applications

Abstract

The non-central negative binomial distribution is both a mixed and compound Poisson distribution with applications in photon and neural counting, statistical optics, astronomy and a stochastic reversible counter system. In this paper various important probabilistic properties of the non-central negative binomial distribution in practical applications like log-concavity, discrete self-decomposability, unimodality, asymptotic behavior and tail length of the probability distribution have been derived. The construction as a mixed Poisson process by specifying a joint distribution for the inter-arrival times and its application is illustrated by a fit to real life data set.     

Uniform stochastic orderings and total positivity

Uniform stochastic orderings of random variables are expressed as total positivity (TP) of density, survival, and distribution functions. The orderings are called uniform because each is a stochastic order that persists under conditioning to a family of intervals—for example, the family consisting of all intervals of the form (-∞,x]. This paper is concerned with the preservation of uniform stochastic ordering under convolution, mixing, and the formation of coherent systems. A general TP₂ result involving preservation of total positivity under integration is presented and applied to convolutions and mixtures of distribution and survival functions. Log-concavity of distribution, survival, and density functions characterizes distributions that preserve the various orderings under convolution. Likewise, distributions that preserve orderings under mixing are characterized by TP₂ distribution and survival functions.     

A discrete analog of Gumbel distribution: properties,parameter estimation and applications

A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.     

GLOBAL LOG-CONCAVITY OF THE LIKELIHOOD IN MODELS WITH GROUPED DISCRETE DATA

The paper considers the property of global log-concavity of the likelihood function in discrete data models in which the data are observed in ‘grouped’ form, meaning that for some observations, while the actual value is unknown, the realisation of the discrete random variable is known to fall within a certain range of values. A typical likelihood contribution in this type of model is a sum of probabilities over a range of realisations. An important issue is whether the property of log-concavity in the ungrouped case carries over to the grouped counterpart; the paper finds, by way of a simple but relevant counter-example, that this is not always the case. However, in two cases of practical interest, namely the Poisson and geometric models, the property of log-concavity is preserved under grouping.     

Uniform stochastic ordering and related inequalities

Stochastic order between univariate random variates may be called uniform when such order persists under conditioning to a broad family of intervals. The ordering is local when it holds for any finite interval (a, b), however small. Local order in multivariate settings has been described by Whitt (1980, 1981), by Karlin and Rinott (1980), and by others. The prevalence of uniform and local order in a variety of simple stochastic-process settings is displayed, and inequalities arising from such orderings developed.