Discrete multivariate distributions (original) (raw)
A family of multivariate discrete distributions
South African Statistical Journal, 2020
In this short paper, the multivariate Poisson-Gamma, the multinomial N-mixture and the negative multinomial distributions are shown to have probability mass functions of the same form and thus to share, broadly, the same distributional properties. The three distributions are, however, fundamentally very different in nature, that is, in terms of genesis, interpretation and model building, and these differences are highlighted and discussed.
A unified approach to bivariate discrete distributions
Metrika, 2007
Here we introduce a bivariate generalized hypergeometric factorial moment distribution (BGHFMD) through its probability generating function (p.g.f.) whose marginal distributions are the generalized hypergeometric factorial moment distributions introduced by Kemp and Kemp (Bull Int Stat Inst 43:336-338,1969). Well-known bivariate versions of distributions such as binomial, negative binomial and Poisson are special cases of this distribution. A genesis of the distribution and explicit closed form expressions for the probability mass function of the BGHFMD, its factorial moments and the p.g.f.'s of its conditional distributions are derived here. Certain recurrence relations for probabilities, moments and factorial moments of the bivariate distribution are also established.
A Generalization to the Family of Discrete Distributions
An alternative approaches for a couple of discrete distributions like Binomial and Multinomial, Poisson, etc having more general form of sampling method (more than one outcome in one trial) compared to tradition sampling heuristics have been suggested and termed as Generalized Binomial, Generalized Multinomial, Generalized Poisson, Generalized Geometric respectively. It is evident that the traditional existing distributions are the special cases of the proposed generalized distributions. The basic distributional properties of the proposed distributions have also been examined including the limiting form. Real life examples are cited for the respective distributions.
Some representations of the multivariate Bernoulli and binomial distributions
Journal of Multivariate Analysis, 1990
Multivariate but vectorized versions for Bernoulli and binomial distributions are established using the concept of Kronecker product from matrix calculus. The multivariate Bernoulli distribution entails a parameterized model, that provides an alternative to the traditional log-linear model for binary variables.
On Some Discrete Distributions and their Applications with Real Life Data
Journal of Modern Applied Statistical Methods, 2009
This article reviews some useful discrete models and compares their performance in terms of the high frequency of zeroes, which is observed in many discrete data (e.g., motor crash, earthquake, strike data, etc.). A simulation study is conducted to determine how commonly used discrete models (such as the binomial, Poisson, negative binomial, zero-inflated and zero-truncated models) behave if excess zeroes are present in the data. Results indicate that the negative binomial model and the ZIP model are better able to capture the effect of excess zeroes. Some real-life environmental data are used to illustrate the performance of the proposed models.
The Discrete Poisson-Garima Distribution
Biometrics & Biostatistics International Journal
The graphs of the pmf of Poisson-Garima distribution (PGD) and Poisson-Lindley distribution (PLD) for varying values of the parameter are shown in the figure 1
A Review on Some New Bivariate and Univariate Discrete Distributions
Bulletin of Faculty of Science, Zagazig University
A three-parameter new discrete analog of Alpha-power Weibull distribution (DAPW) is presented. Some of its fundamental distributional and reliability properties are established. Two datasets are used showing the flexibility of the proposed model. An attempt to introduce a new lifetime model as a discrete version of the continuous exponentiated exponential distribution which is called discrete exponentiated exponential distribution (DEE). a discrete analog of the continuous generalized inverted exponential distribution denoted by discrete generalized inverted exponential distribution (DGIE). The model with two real data sets is also examined. In addition, the developed DGIE is applied as color image segmentation which aims to cluster the pixels into their groups. To evaluate the performance of DGIE, a set of six color images is used, as well as it is compared with other image segmentation methods including Gaussian mixture model (GMM), K-means, and Fuzzy subspace clustering (FSC). The DGIE provides high performance than other competitive methods.
On a General Class of Discrete Bivariate Distributions
Sankhya B
In this paper we develop a very general class of bivariate discrete distributions. The basic idea is very simple. The marginals are obtained by taking the random geometric sum of a baseline distribution function. The proposed class of distributions is a very flexible class of distributions in the sense the marginals can take variety of shapes. It can be multimodal as well as heavy tailed also. It can be both over dispersed as well as under dispersed. Moreover, the correlation can be of a wide range. We discuss different properties of the proposes class of bivariate distributions. The proposed distribution has some interesting physical interpretations also. Further, we consider two specific base line distributions namely; Poisson and negative binomial distributions for illustrative purposes. Both of them are infinitely divisible. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form. They can be obtained by solving three and five dimensional non-linear optimizations problems, respectively. To avoid that we propose to use the method of moment estimators and they can be obtained quite conveniently. The analyses of two real data sets have been performed to show the effectiveness of the proposed class of models. Finally, we discuss some open problems and conclude the paper.
In this work the class of discrete probabilistic distributions of the II-nd type of random sets of event is investigated. As the tool for constructing of such probabilistic distributions it is offered to use associative functions. There is stated a new approach to define a discrete probabilistic distribution of the II-nd type of a random set on a finite set of N events on the basis of obtained recurrence relation and a given associative function. Advantage of the offered approach is that for definition of probabilistic distribution instead of a totality of 2 N probabilities it is enough to know N probabilities of events and a type of associative function. In this paper an |X|-ary covariance of a random set of events is considered. It is a measure of the additive deviation of the events from the independent situation. The process of recurrent constructing a probabilistic distribution II-nd type is demonstrated by the example of three associative functions. The proof of the legitimacy / illegitimacy the obtained distribution by passing to the probabilistic distribution of the I-st type by formulas of Möbius is given. Theorems that establish the form and conditions of the legitimacy of the resulting probabilistic distributions are proved. |X|-ary covariances of random sets of events are found.