On estimation of the PDF and the CDF of the one-parameter polynomial exponential family of distributions

Abstract

In this article, we have considered the estimation of the probability density function and cumulative distribution function of the one-parameter polynomial exponential family of distributions. A number of probability distributions like the exponential, Lindley, length-biased Lindley and Sujatha are particular cases. Two estimators—maximum likelihood and uniformly minimum variance unbiased estimators of the probability density function and cumulative distribution function of the family have been discussed. The estimation issues of the length-biased Lindley and Sujatha distribution have been considered in detail. The estimators have been compared in mean squared error sense. Monte Carlo simulations and real data analysis are performed to compare the performances of the proposed estimators.

Keywords:

• Lindley distribution
• maximum likelihood estimator
• uniformly minimum variance unbiased estimator

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 62F10

Acknowledgement

The authors would like to thank the referee for valuable comments and suggestions that have encouraged the authors to make significant improvements in the article. The first author acknowledges the Department of Science and Technology, India (IF150324), for sponsoring the research work in term of providing Fellowship.