Compound zero-truncated Poisson normal distribution and its applications

Abstract

Here, we first propose three-parameter model and call it as the compound zero-truncated Poisson normal (ZTP-N) distribution. The model is based on the random sum of N independent Gaussian random variables, where N is a zero truncated Poisson random variable. The proposed ZTP-N distribution is a very flexible probability distribution function. The probability density function can take variety of shapes. It can be both positively and negatively skewed, moreover, normal distribution can be obtained as a special case. It can be unimodal, bimodal as well as multimodal also. It has three parameters. An efficient EM type algorithm has been proposed to compute the maximum likelihood estimators of the unknown parameters. We further propose a four-parameter bivariate distribution with continuous and discrete marginals, and discuss estimation of unknown parameters based on the proposed EM type algorithm. Some simulation experiments have been performed to see the effectiveness of the proposed EM type algorithm, and one real data set has been analyzed for illustrative purposes.

Keywords:

• Poisson distribution
• skew normal distribution
• maximum likelihood estimators
• EM type algorithm
• Fisher information matrix

MATHEMATICS SUBJECT CLASSIFICATION:

• 62E15
• 62F10
• 62H10

Acknowledgments

The authors would like to thank unknown reviewers for their constructive suggestions which have helped us to improve the manuscript.

Additional information

Funding

This research project was funded by the Deanship of Scientific Research (DSR), The University of Jordan. The first author, therefore, acknowledges with thanks DSR financial support. Part of the work of the second author has been funded by a research grant from the Science and Engineering Research Board, Government of India.