Non-central Negative Binomial Distribution Research Articles

Sequential probability ratio test for zero inflation in counting data

In this paper, we will discuss the sequential probability ratio test to test the zero inflation in counting data with the sample size being a random variable with specified Type I and Type II errors. We will introduce the zero-inflated non-central negative binomial distribution, and explain how its operating characteristic and average sample number functions are derived. Simulations are studied to indicate the performances of the proposed tests with different scenarios. A real application is also discussed to explain the process of the sequential probability ratio test.

COUNTING DISTRIBUTIONS IN RISK THEORY

In this paper we introduce some significant counting distributionsin risk theory. The first one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

COUNTING DISTRIBUTIONS IN RISK THEORY

In this paper we introduce some significant counting distributionsin risk theory. The first one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

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

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.

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.

Generalizations of Non Central Negative Binomial, Charlier Series Distributions, and Their Extensions by Lagrange Expansion

The non central negative binomial distribution (NNBD), the Charlier series distribution (CSD), and some of their recent extensions are formulated as Lagrange probability distributions of the second kind. Derivation of Lagrangian NNBD, CSD, and their extensions are also considered. Some formulations of the proposed new distributions are given. Moments, index of dispersion, parameter estimation and examples of data fitting for the Lagrangian NNBD are presented.

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.

Some generating functions for the Laguerre and related polynomials

The authors investigate (and present generalizations of) several generating functions of the classical Laguerre and related polynomials. One of these generating functions was recently rederived probabilistically by Lee [P.-A. Lee, Bull. Inst. Math. Acad. Sinica 25 (1997) 151–154], who made use of the theory of a class of probability distributions known as the non-central negative binomial distribution. Basic (or q-) extensions of the main series identities are also considered.

Random deletion model and Rainville's generating function for the laguerre polynomials

A random deletion model associated with the non-central negative binomial distribution is used to provide a new proof of the well-known generating function formula of Rainville for the generalised Laguerre polynomials. The proof, using only probabilistic arguments, relies on the Poisson mixture representation of the distribution and the commutative nature of the deletion and mixing operations.

Some notes on the non-central negative binomial distribution

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.

The bivariate non-central negative binomial distributions

Four bivariate generalisations (Type I–IV) of the non-central negative binomial distribution (Ong/Lee) are considered. The Type I generalisation is constructed using the “latent structure model” scheme (Goodman) while the Type II generalisation arises from a variation of this scheme. The Type III generalisation is formed by using the method of random elements in common (Mardia). The Type IV is an extension of the Type I generalisation. Properties of these bivariate distributions including joint central and factorial moments are discussed; several recurrence formulae of the probabilities are given. An application to the childhood accident data of Mellinger et al. is considered with the precision of the Type I maximum likelihood estimates computed.

Bivariate non-central negative binomial distribution: Another generalisation

Another bivariate generalisation (Type V) of the non-central negative binomial distribution is considered. This generalisation is constructed (i) as a latent structure model; (ii) as an extension of an accident proneness model investigated by Edwards/Gurland (1961); and (iii) as a reversible stochastic counter model. The third construction gives, as a result, an apparently new formulation of the Edwards/Gurland model. The probabilities, moments, recurrence formulas and some properties are given. An application to the data used by Holgate (1966) is considered.

Neural Counting and Photon Counting in the Presence of Dead Time

The usual stimulus-based neural counting model for audition is found to be mathematically identical to the well-know semiclassical formalism for photon counting. In particular, we explicitly demonstrate the equivalence of McGill's noncentral negative binomial distribution and Pe\ifmmode \check{r}\else \v{r}\fi{}ina's multimode confluent hypergeometric distribution for a coherent signal imbedded in chaotic noise. Dead-time corrections, important both in neural counting and in photon counting, are incorporated in a generalized form of this distribution. Some specific implications of these results are discussed.