Efficient Estimator of Parameters of a Multivariate Geometric Distribution

S. Annapurna,U. J. Dixit

doi:10.2991/jsta.2018.17.4.6

Abstract

The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of a multivariate geometric distribution (MGD) have been derived. A modification of the MLE estimator (modified MLE) has been derived in which case the bias is reduced. The mean square error (MSE) of the modified MLE is less than the MSE of the MLE. Variances of the parameters and the corresponding generalized variance (GV) has been obtained. It has been shown that the MLE and modified MLE are consistent estimators. A comparison of the GVs of modified MLE and UMVUE has shown that the modified MLE is more efficient than the UMVUE. In the final section its application has been discussed with an example of actual data.

Highlights

It is appropriate and convenient to measure lifetime of devices such as on/off switches, bulbs, engines of an airplane on a discrete scale
Dixit and Annapurna [4] have further obtained the uniformly minimum variance unbiased estimator (UMVUE) estimators and have compared the maximum likelihood estimator (MLE) and UMVUE based on the (MSEs)
We have further shown that this modified MLE is better than the UMVUE

Summary

INTRODUCTION

It is appropriate and convenient to measure lifetime of devices such as on/off switches, bulbs, engines of an airplane on a discrete scale. Various models of the bivariate geometric distribution (BGD) have been proposed to study lifetime devices. Hare Krishna and Pundir [3] have obtained the MLE and Bayes estimators of the parameters for this BGD. Dixit and Annapurna [4] have further obtained the UMVUE estimators and have compared the MLE and UMVUE based on the (MSEs). In this paper we obtain UMVUE and MLE of the parameters in Eq (1) and their functions. We look at another very popular principle used namely method of maximum likelihood to obtain the estimators of the functions of the parameters. These shall be compared with the corresponding estimators obtained by UMVUE in order to study their efficiency.

CONSISTENCY OF MODIFIED MLE

CONCLUSION AND COMPARISION OF