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.     

Some reliability characteristics and stochastic ordering of series and parallel systems of bivariate generalized exponential distribution

In this paper, the reliability properties of two-component parallel and series systems are considered for bivariate generalized exponential (BVGE) distribution introduced by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivar Anal. 2009;100:581–593]. For this model, the moments and mean residual life functions of these systems and the regression mean residual life function are derived. Stochastic comparisons of series and parallel systems of BVGE distribution are investigated. Moreover, various ordering results for the comparisons of series and parallel systems arising from BVGE random vectors are obtained with respect to the usual stochastic, reversed hazard rate and likelihood ratio orderings.     

Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

On some generalized ageing orderings

Some partial orderings that compare probability distributions with the exponential distribution are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which describe the same kind of phenomenon of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.     

A COM-Poisson-type generalization of the negative binomial distribution

ABSTRACT

This paper introduces a generalization of the negative binomial (NB) distribution in analogy with the COM-Poisson distribution. Many well-known distributions are particular and limiting distributions. The proposed distribution belongs to the modified power series, generalized hypergeometric and exponential families, and also arises as weighted NB and COM-Poisson distributions. Probability and moment recurrence formulae, and probabilistic and reliability properties have been derived. With the flexibility to model under-, equi- and over-dispersion, and its various interesting properties, this NB generalization will be a useful model for count data. An application to empirical modeling is illustrated with a real data set.     

A family of skew-symmetric-Laplace distributions

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we shall introduce another more general class of skew distributions, specially related to the Laplace distribution. This new class contains some previously known skew distributions. We shall investigate different characteristics of members of this class such as its moments, thus generalizing a result of Umbach (Stat Probab Lett 76:507?C512, 2006), limiting behavior, moment generating function, unimodality and reveal its natural occurrence as the distribution of some order statistics. In addition, we will generalize a result of Aryal and Rao (Nonlinear Anal 63:639?C646, 2005) in connection with truncated skew-Laplace distribution and study its certain stochastic orderings. Some illustrative examples are also provided.     

A Proportional Hazard Marshall–Olkin Extended Family of Distributions and its Application to Gompertz Distribution

In this paper, we have presented a proportional hazard version of the Marshall–Olkin extended family of distributions. This family of distributions has been compared in terms of stochastic orderings with the Marshall-Olkin extended family of distributions. Considering the Gompertz distribution as the baseline, the monotonicity of the resulting failure rate is shown to be either increasing or bathtub, even though the Gompertz distribution has an increasing failure rate. The maximum likelihood estimation of the parameters has been studied and a data set, involving the serum–reversal times, has been analyzed and it has been shown that the model presented in this paper fit better than the Gompertz or even the Mrashall–Olkin Gompertz distribution. The extension presented in this paper can be used in other family of distributions as well.     

On some generalized orderings: In the spirit of relative ageing

ABSTRACT

We introduce some new generalized stochastic orderings (in the spirit of relative ageing) which compare probability distributions with the exponential distribution. These orderings are useful to understand the phenomenon of positive ageing classes and also helpful to guide the practitioners when there are crossing hazard rates and/or crossing mean residual lives. We study some characterizations of these orderings. Inter-relations among these orderings have also been discussed.     

Inflated modified power series distributions with applications

In certain applications involving discrete data, it is sometimes found that X = 0 is observed with a frequency significantly higher than predicted by the assumed model. Zero inflated Poisson, binomial and negative binomial models have been employed in some clinical trials and in some regression analysis problems.

In this paper, we study the zero inflated modified power series distributions (IMPSD) which include among others the generalized Poisson and the generalized negative binomial distributions and hence the Poisson, binomial and negative binomial distributions. The structural properties along with the distribution of the sum of independent IMPSD variables are studied. The maximum likelihood estimation of the parameters of the model is examined and the variance-covariance matrix of the estimators is obtained. Finally, examples are presented for the generalized Poisson distribution to illustrate the results.     

Reliability study of proportional odds family of discrete distributions

The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall–Olkin family of distributions or Marshall–Olkin extended distributions. In this paper, we consider Marshall–Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples.     

Flexible Latent‐State Modelling of Old Faithful's Eruption Inter‐Arrival Times in 2009

This paper is concerned with the analysis of a time series comprising the eruption inter‐arrival times of the Old Faithful geyser in 2009. The series is much longer than other well‐documented ones and thus gives a more comprehensive insight into the dynamics of the geyser. Basic hidden Markov models with gamma state‐dependent distributions and several extensions are implemented. In order to better capture the stochastic dynamics exhibited by Old Faithful, the different non‐standard models under consideration seek to increase the flexibility of the basic models in various ways: (i) by allowing non‐geometric distributions for the times spent in the different states; (ii) by increasing the memory of the underlying Markov chain, with or without assuming additional structure implied by mixture transition distribution models; and (iii) by incorporating feedback from the observation process on the latent process. In each case it is shown how the likelihood can be formulated as a matrix product which can be conveniently maximized numerically.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

A STOCHASTIC ORDERING FOR RANDOM VARIABLES WITH APPLICATIONS

A new univariate stochastic ordering is introduced. Some characterization results for such an ordering are stated. It is proved that the ordering is an integral stochastic ordering, obtaining a maximal generator. By means of this generator, the main properties of the ordering are deduced. A method for introducing univariate stochastic orderings, suggested by the new ordering, is analysed. Relationships with other stochastic orderings are also developed. To conclude, an example of an application of the new ordering to the field of medicine is proposed.     

Kurtosis and spread

An increase in kurtosis is achieved through the location- and scale-free movement of probability mass from the “shoulders” of a distribution into its centre and tails. We introduce a coherent structure of ordering and measures, requiring no symmetry assumption, that represent different formalizations of this movement. For this purpose spread functions and spread-spread plots are defined. The orderings impose growth patterns on the spread-spread plot of the distributions involved, and the weakest involve both a specific scale-matching technique and placement of “shoulders”. The role of existing kurtosis orderings and measures in this general context is identified and examples discussed throughout.     

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.     

Order statistics and the length of the best-n-of-(2n − 1) competition

The best-4-of-7 series is a popular playoff format to decide the champion in most North American professional sports. World Series (best-4-of-7) type competitions give rise to interesting probabilistic and statistical questions. We determine the expected length of this type of series by relating it to a problem involving order statistics. We also calculate the variance of the length and provide a simple formula for series of fair games. The method can be extended to derive higher order moments. This novel approach leads to new results that can be formulated in closed forms in terms of the distribution function of various binomial distributions. The emphasis is on establishing the connection to order statistics and obtaining closed forms. The relation to the negative binomial distribution as well as to the sooner waiting time problem in sequential testing is also discussed. We also consider the case when ties are allowed in the single games.     

S–S method for stochastically ordered multinomial populations with missing data

Stochastic ordering is a useful concept in order restricted inferences. In this paper, we propose a new estimation technique for the parameters in two multinomial populations under stochastic orderings when missing data are present. In comparison with traditional maximum likelihood estimation method, our new method can guarantee the uniqueness of the maximum of the likelihood function. Furthermore, it does not depend on the choice of initial values for the parameters in contrast to the EM algorithm. Finally, we give the asymptotic distributions of the likelihood ratio statistics based on the new estimation method.     

On some distributions arising in inverse cluster sampling

In this paper, distributions of items sampled inversely in clusters are derived. In particular, negative binomial type of distributions are obtained and their properties are studied. A logarithmic series type of distribution is also defined as a limiting form of the obtained generalized negative binomial distribution.     

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.     

Testing for an excessive number of zeros in time series of bounded counts

For the modeling of bounded counts, the binomial distribution is a common choice. In applications, however, one often observes an excessive number of zeros and extra-binomial variation, which cannot be explained by a binomial distribution. We propose statistics to evaluate the number of zeros and the dispersion with respect to a binomial model, which is based on the sample binomial index of dispersion and the sample binomial zero index. We apply this index to autocorrelated counts generated by a binomial autoregressive process of order one, which also includes the special case of independent and identically (i. i. d.) bounded counts. The limiting null distributions of the proposed test statistics are derived. A Monte-Carlo study evaluates their size and power under various alternatives. Finally, we present two real-data applications as well as the derivation of effective sample sizes to illustrate the proposed methodology.