A Generalised Exponential-Lindley Mixture of Poisson Distribution

Abstract

Background: The exponential and the Lindley (1958) distributions occupy central places among the class of continuous probability distributions and play important roles in statistical theory. A Generalised Exponential-Lindley Distribution (GELD) was given by Mishra and Sah (2015) of which, both the exponential and the Lindley distributions are the particular cases. Mixtures of distributions form an important class of distributions in the domain of probability distributions. A mixture distribution arises when some or all the parameters in a probability function vary according to certain probability law. In this paper, a Generalised Exponential- Lindley Mixture of Poisson Distribution (GELMPD) has been obtained by mixing Poisson distribution with the GELD.
 Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution.
 Results: The Probability mass of function of generalized exponential-Lindley mixture of Poisson distribution has been obtained by mixing Poisson distribution with GELD. The first four moments about origin of this distribution have been obtained. The estimation of its parameters has been discussed using method of moments and also as maximum likelihood method. This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature and it has been observed that the distribution gives a better fit than the Poisson–Lindley Distribution (PLD) of Sankaran (1970).
 Conclusion: P-value of the GELMPD is found greater than that in case of PLD. Hence, it is expected to be a better alternative to the PLD of Sankaran for similar type of discrete data-set which is negative binomial in nature.

On a Generalised Exponential-Lindley Mixture of Generalised Poisson Distribution

“…This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature. P-value of this distribution has been compared to the PLD of Sankaran (1970) and GELMPD of Sah and Mishra (2019) for similar type of data-sets and finally, conclusions have been drawn.…”

“…It is reduced to the Poisson -Lindley distribution of Sankaran (1970) at  = 1. The first four moments about origin of the GELMPD have been obtained as ( 4 3 2 4 5 24 4 36 3 14 2 … … (8) The equations (4) to (8) were obtained by Sah and Mishra (2019). Consul and Jain (1973) obtained a two-parameter generalised Poisson distribution (GPD) given by its probability function…”

An Alternative Count Distribution for Modeling Dispersed Observations

“…The data was earlier used by Frees (2010) and Frees and Valdez (2008). The fourth dataset is the frequency of mistakes in copying groups of random digits first reported by Kemp and Kemp (1965) and in several other propositions (Sah & Mishra, 2019;Samutwachirawong, 2021;Shanker & Mishra, 2014;Shanker & Mishra, 2016) involving new discrete distributions. The fifth dataset (yeast cell counts per square) was used to assess the mixed Poisson Lindley distribution and other related distributions (Shanker & Hagos, 2015).…”

Modelling Count Variables: A Comparative Analysis of two Discretization Techniques

“…The third dataset is the frequency of epileptic seizures previously used on other discrete distributions [25,57]. The fourth dataset on the number of mistakes in copying groups of random digits has been used to model various count distributions [25,6,28,58]. The last dataset represents the number of strikes in a UK coal mining industries from 1948-1959 [16,59].…”