ON HYPERGEOMETRIC GENERALIZED NEGATIVE BINOMIAL DISTRIBUTION

It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.


Introduction.
A certain mixture distribution arises when all (or some) parameters of a distribution vary according to some probability distribution, called the mixing distribution.A well-known example of discrete-type mixture distribution is the negative binomial distribution which can be obtained as a Poisson mixture with gamma mixing distribution.
We use the notation HGNB(α,a,p) to denote the hypergeometric generalized negative binomial distribution with pmf (1.6).
(ii) If a = 2, p = 1, then, see the appendix, using the relation which is the pmf of Poisson-Lindley distribution with parameter α considered by Sankaran [7].
The negative binomial distribution provides a more flexible alternative to the Poisson distribution particularly when the variance of the data is significantly larger than the mean.Johnson et al., [4,Chapter 5], provides a comprehensive survey of the applications and generalizations/extensions of the negative binomial distributions.
The discrete Poisson-Lindley distribution was shown by Sankaran [7] to provide, for particular data sets, better fit than other discrete distributions such as negative binomial, Poisson and Hermite distributions.Yet, no attempt has been made to study the properties of this distribution analytically.
The aim of this paper is to investigate some important properties of the hypergeometric generalized negative binomial distribution.These include existence of moments as well as properties of statistical measures such as the index of dispersion, skewness, and kurtosis.Also, a recurrence relation for calculating probabilities from the considered distribution is given.Finally, the distribution is fitted to entomological field data and its goodness-of-fit is demonstrated.

Moments and associated measures.
We start this section by showing that the HGNB distribution has moments of all positive orders.Theorem 2.1.For all α, a, p > 0, the HGNB(α,a,p) distribution has moments of all positive orders where S(r , n) are the Stirling numbers of the second kind Hence, the factorial moments of X, that is, are given by Making use of the following integral, see Erdély [3], provided Re b, Re s > 0, Re s > Re q, |s| > |q|, we obtain (2.6) Now using [1, formula (15.3.4),page 559]: , and the definition of hypergeometric function, respectively, we obtain which is the r th moment of the Poisson-Lindley distribution with pmf (1.11).
(iii) If a = 1, p = 2, then, see the appendix, using the relation which is the r th moment of the generalized mixture of geometric distributions with pmf (1.13).
Proof.The characteristic function of X ∼ HGNB(α,a,p), see [2, page 28], is given by Using the cumulant generating function K X (t) = ln[ψ X (t)], the r th cumulant of X is given by κ r = i −r (d r /dt r )K X (0).Therefore, the first four cumulants of X, respectively, are given by (2.14) Recall that the index of dispersion (ID), skewness ( β 1 ), and kurtosis (β 2 ) of X, respectively, are given by (2.15) > 0, proving the theorem.
Remarks.(i) If a = p, the index of dispersion does not depend on a while the skewness and kurtosis depend on a.

Application. Table
In calculating the chi-square statistic 2 /e i , where o i (e i ) are the observed (expected) frequencies, m = 13 after combining the observed (expected) frequencies corresponding to counts 12 to 25, as did McGuire et al. [5], that is, o 13 = 15 and e 13 = 14.87(14.55)for NB (HGNB).Also, the degrees of freedom are given by m − t − 1 where t = 2(3) is the number of estimated parameters for NB (HGNB).
From Table 4.2, we observe that fitting the HGNB distribution gives an improvement over fitting the NB distribution as judged by the chi-square value.
Proof of (1.10).Using (A.1) with a = 2, c = 1, and the definition of the hypergeometric function, respectively, we obtain

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.