Estimating Common Scale Parameter of Two Logistic Populations: A Bayesian Study

Abstract

Estimation under equality restrictions is an age old problem and has been considered by several researchers in the past due to practical applications and theoretical challenges involved in it. Particularly, the problem has been extensively studied from classical as well as decision theoretic point of view when the underlying distribution is normal. In this paper, we consider the problem when the underlying distribution is non-normal, say, logistic. Specifically, estimation of the common scale parameter of two logistic populations has been considered when the location parameters are unknown. It is observed that closed forms of the maximum likelihood estimators (MLEs) for the associated parameters do not exist. Using certain numerical techniques the MLEs have been derived. The asymptotic confidence intervals have been derived numerically too, as these also depend on the MLEs. Approximate Bayes estimators are proposed using non-informative as well as conjugate priors with respect to the squared error (SE) and the LINEX loss functions. A simulation study has been conducted to evaluate the proposed estimators and compare their performances through mean squared error (MSE) and bias. Finally, two real life examples have been considered in order to show the potential applications of the proposed model and illustrate the method of estimation.

KEYWORDS AND PHRASES:

• Approximate Bayes estimator
• asymptotic confidence interval
• bias of an estimator
• Lindley’s approximation
• LINEX loss function
• maximum likelihood estimator (MLE)
• mean squared error (MSE)

Acknowledgment

The authors would like to express their sincere thanks to an anonymous reviewer and the editor for their constructive suggestions and comments which have helped significantly in improving the presentation of this work. The Second author (Manas Ranjan Tripathy) would like to thank Science and Engineering Research Board (SERB), EMR/2017/003078, Department of Science and Technology (DST), New Delhi, India for providing some financial support.

Additional information

Funding

Science and Engineering Research Board.