The negative binomial distribution has become increasingly popular as a more flexible alternative to Poisson distribution, especially when it is questionable whether the strict requirements for Poisson distribution could be satisfied. But negative binomial distribution is better for overdispersed count data that are not necessarily heavy-tailed, for heavy tailed count data the traditional stati...

We assume that product i has a constant hazard of adjusting, fi, in each month, where fi ∼ Beta(a, b). Let us denote the product’s lifetime by ni. These assumptions imply that the total number of price changes in a product’s lifetime is distributed according to the binomial distribution, xi ∼ Bin(ni, fi). We assume, furthermore, that fi is distributed according to the beta distribution, fi ∼ Be...

background: in the case of sensitive questions such as number of alcoholics known, majority of respondents might give an answer of zero. poisson regression model (p) is the standard tool to analyze count data. however, p provides poor fit in the case of zero inflated counts, when over-dispersion exists. therefore, the questions to be addressed are to compare performance of alternative count reg...

Based on the probability generating function of stuttering Poisson distribution (SPD), this paper considers some equivalent propositions of SPD. From this, we show that some distributions in the application of non-life insurance actuarial science are SPD, such as negative binomial distribution, compound Poisson distribution etc.. By weakening condition of equivalent propositions of SPD, we defi...

Consider k success counts from a binomial distribution with unknown N and success probability p. We consider the problem of estimating N. By integrating the likelihood for Nand p over a beta density for p, we obtain the beta-binomial distribution which results in stable and reasonably efficient estimators of N, which compare favorably with and are often better than the estimates introduced by O...

Let F q X] denote a polynomial ring over a nite eld F q with q elements. Let P n be the set of monic polynomials over F q of degree n. Assuming that each of the q n possible monic polynomials in P n is equally likely, we give a complete characterization of the limiting behavior of P((n = m) as n ! 1 by a uniform asymptotic formula valid for m 1 and n?m ! 1, where n represents the number (multip...

We investigate the KNO scaling function of the modified negative binomial distribution (MNBD), because this MNBD can explain the oscillating behaviors of the cumulant moment observed in e+e− annihilations and in hadronic collisions. By using a straightforward method and the Poisson transform we derive the KNO scaling function from the MNBD. The KNO form of experimental data in e+e− collisions a...

Fractional derivative in time variable is introduced into the Fokker-Planck equation of a population growth model. It’s solution, the KNO scaling function, is transformed into the generating function for the multiplicity distribution. Formulas of the factorial moment and the Hj moment are derived from the generating function, which reduces to that of the negative binomial distribution (NBD), if...

In comparing the mean count of two independent samples, some practitioners would use the t-test or the Wilcoxon rank sum test while others may use methods based on a Poisson model. It is not uncommon to encounter count data that exhibit overdispersion where the Poisson model is no longer appropriate. This paper deals with methods for overdispersed data using the negative binomial distribution r...

Problem statement: The modeling of claims count is one of the most important topics in actuarial theory and practice. Many attempts were implemented in expanding the classes of mixed and compound distributions, especially in the distribution of exponential family, resulting in a better fit on count data. In some cases, it is proven that mixed distributions, in particular mixed Poisson and mixed...