Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions

Abstract

In this paper, we derive Bayesian estimators of the parameters of modified power series distributions inflated at any of a support point under linex and general entropy loss function. We assume that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution inflated at a given point.     

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.     

Bayesian Estimation for Deformed Modified Power Series Distribution

In this article, posterior distribution, posterior moments, and predictive distribution for the modified power series distributions deformed at any of a support point under linex and generalized entropy loss function are derived. It is assumed that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma, or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution deformed at a given point.     

A Family Which Integrates the Generalized Charlier Series and Extended Noncentral Negative Binomial Distributions

The generalized Charlier series distribution includes the binomial distribution, and the noncentral negative binomial distribution extends the negative binomial distribution. The present article proposes a family of counting distributions, which contains both the generalized Charlier series and extended noncentral negative binomial distributions. Compound and mixture formulations of the proposed distribution are given. The probability mass function is expressible in terms of the confluent hypergeometric function as well as the Gauss hypergeometric function. Recursive formulae for probability mass function have been studied by Panjer, Sundt and Jewell, Schröter, Sundt, and Kitano et al. in the context of insurance risk. This article explores horizontal, vertical, triangular, and diagonal recursions. Recursive formulae as well as exact expressions for descending factorial moments are studied. The proposed distribution allows overdispersion or underdispersion relative to a Poisson distribution. An illustrative example of data fitting is given.     

Estimation of probabilities in the class of modified power series distributions

In this paper we consider the class of modified power series distribution introduced by GUPTA (1974) and derive a minimum variance unbiased estimator (MVUE) of the probability function for this class. these results are then applied to obtain MVUE of the probability function for the generalized negative binomial distributions, the generalized poisson distribution, the generalized logarithmic series distribution and the lost game distribution. A large number of results in the literature follow trivially from out results as special cases.     

Asymptotic Normality in Mixtures of Power Series Distributions

Abstract.  The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.     

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θ^r (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

Families of power series distributions,with particular reference to the Lerch family

The paper revisits the concept of a power series distribution by defining its series function, its power parameter, and hence its probability generating function. Realization that the series function for a particular distribution is a special case of a recognized mathematical function enables distributions to be classified into families. Examples are the generalized hypergeometric family and the q-series family, both of which contain generalizations of the geometric distribution. The Lerch function (a third generalization of the geometric series) is the series function for the Lerch family. A list of distributions belonging to the Lerch family is provided.     

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.     

Semiparametric inference based on a class of zero-altered distributions

In modeling count data collected from manufacturing processes, economic series, disease outbreaks and ecological surveys, there are usually a relatively large or small number of zeros compared to positive counts. Such low or high frequencies of zero counts often require the use of underdispersed or overdispersed probability models for the underlying data generating mechanism. The commonly used models such as generalized or zero-inflated Poisson distributions are parametric and can usually account for only the overdispersion, but such distributions are often found to be inadequate in modeling underdispersion because of the need for awkward parameter or support restrictions. This article introduces a flexible class of semiparametric zero-altered models which account for both underdispersion and overdispersion and includes other familiar models such as those mentioned above as special cases. Consistency and asymptotic normality of the estimator of the dispersion parameter are derived under general conditions. Numerical support for the performance of the proposed method of inference is presented for the case of common discrete distributions.     

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.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

On the Bayes estimators of the parameters of size-biased generalized power series distributions

ABSTRACT

In this paper, we derive the Bayes estimators of functions of parameters of the size-biased generalized power series distribution under squared error loss function and weighted square error loss function. The results of size-biased GPSD are then used to obtain particular cases of the size-biased negative binomial, size-biased logarithmic series, and size-biased Poisson distributions. These estimators are better than the classical minimum variance unbiased estimators in the sense that they increase the range of the estimation. Finally, an example is provided to illustrate the results and a goodness of fit test is done using the maximum likelihood and Bayes estimators.     

Conditional maximum likelihood estimation in negative binomial INGARCH processes with known number of successes when the true parameter is at the boundary of the parameter space

The paper establishes the asymptotic distribution of the conditional maximum likelihood estimator for integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes of conditional negative binomial distributions, with the number of successes in the definition of the negative binomial distribution being assumed to be known, when the true parameter is at the boundary of the parameter space. Based on the result, coefficient nullity tests are developed for model simplification. The proposed tests are investigated through a simulation study.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.     

Modeling spatial overdispersion via the generalized Waring point process

Modeling spatial overdispersion requires point process models with finite‐dimensional distributions that are overdisperse relative to the Poisson distribution. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. In addition, although processes based on negative binomial finite‐dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we change perspective and construct a new process based on a different overdisperse count model, namely, the generalized Waring (GW) distribution. While comparably tractable and flexible to negative binomial processes, the GW process is shown to possess all required properties and additionally span the negative binomial and Poisson processes as limiting cases. In this sense, the GW process provides an approximate resolution to the conundrum highlighted by Diggle and Milne.     

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.