On three-parameter generalized exponential distribution

Abstract

In this article, we consider the estimation of the three-parameter generalized exponential distribution. In presence of the unknown location parameter the usual maximum likelihood estimators do not exist for $0<\alpha <1.$ The maximum product of spacings method serves as good alternative since it always exists and yields consistent estimators in the entire parametric space. We develop the asymptotic distribution of the proposed estimators along-with a detailed discussion about the computational intricacies involved in implementing the product of spacing method. Extensive simulations have been performed to demonstrate the effectiveness of the proposed method for α in the range (0, 1), in addition to a comparative study with some standard techniques known to provide consistent estimators, even for $0<\alpha <1.$ Furthermore, two real data sets have been analyzed to demonstrate the applicability of the proposed method.

Keywords:

• Maximum product of spacings
• Non-regular estimable condition
• Percentile bootstrap intervals

Acknowledgements

The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments which have significantly improved the paper. The first author is also thankful for the financial assistance through Institute of Eminence Seed Grant of Banaras Hindu University, Varanasi.

Notes

1 Any value smaller than x₁ can also be used as ${\widehat{\mu }}_0$

2 nlm in R

3 The Q-Q plots for this result has been avoided to prevent superfluous length of the article.

4 (specifically nlminb function)

5 Gupta and Kundu (2001a) reported ${\chi }^2=3.383$

6 Distribution fitted based on MPS estimates.