BAYESIAN ESTIMATIION OF SHAPE PARAMETER OF GENERALIZED INVERSE EXPONENTIAL DISTRIBUTION UNDER THE NON-INFORMATIVE PRIORS

BAYESIAN ESTIMATIION OF SHAPE PARAMETER OF GENERALIZED INVERSE EXPONENTIAL DISTRIBUTION UNDER THE NON-INFORMATIVE PRIORS

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Abstract

In this research, the shape parameter of the Generalized Inverse Exponential Distribution (GIED)

was estimated using maximum likelihood and Bayesian estimation techniques. The Bayes

estimates were obtained under the squared error loss function and precautionary loss function

under the assumption of two non-informative priors. An extensive Monte Carlo simulation study

was carried out to compare the performances of the Bayes estimates with that of the maximum

likelihood estimates at different sample sizes. It was found out that the maximum likelihood have

the same estimate with the Jeffrey’s prior using the squared error loss function, and also

performed better than the Bayes estimates under the Jeffrey’s prior using the precautionary loss

function and uniform prior using both loss function but performed lesser than the Extended

Jeffrey’s prior under both loss functions. The Extended Jeffrey’s prior was observed to have

estimated the shape parameter of the GIED better when compared with the maximum likelihood

estimator and other Bayes estimate at all sample sizes using their mean squared error. Also the

squared error loss function under the Extended Jeffrey’s prior has the best estimate when

compared with other Bayes estimates using their posterior risk. Hence the Bayes estimate under

the Extended Jeffrey’s using the squared error loss function has the best estimator for estimating

the shape parameter of the GIED.

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CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

In the past, many generalized univariate continuous distribution have been proposed. The

generalization of these distributions is important in order to make its shape more flexible to

capture the diversity present in the observed dataset. One of such generalizations is the

Generalized Inverse exponential distribution (GIED) proposed by Abouammoh and Alshangiti

(2009), in which the shape parameter was added to make the distribution more flexible. As a

result, this parameter has to be estimated using the appropriate estimation technique. One of such

techniques is the Bayesian method of estimation which combines the prior knowledge with new

observations to come up with updated information.

Researchers have estimated the parameter of different distributions using the Bayesian technique

because of its advantage over other methods of estimation. Some of this research includes the

work of Farhad et al., (2013) which studied the scale parameter of inverse weibull distribution.

Also, Dey (2015) studied the inverted exponential distribution using this technique.

Although the GIED has been studied using this technique under the assumption of the

informative prior, but there are situations where we do not have information about the prior as

such there will be need to study it under the non-informative prior. It is in the light of this that,

this research intends to study the estimation of the shape parameter of the GIED under the non-

informative priors using two loss functions with the assumption that the scale parameter is

known.

1


1.2 Generalized Inverse Exponential Distribution

One of the simplest and most widely discussed distributions that is used for life testing is the one

parameter exponential distribution. The distribution plays a vital role in the development of

theories. One of the limitations of this distribution is that its applicability is restricted to a

constant hazard rate. This is because there is hardly any system that has time independent hazard

rate. As a result, a number of generalizations of the exponential distribution have been proposed

in earlier literatures, for example the gamma distribution which is sum of independent

exponential variates.

One of the extension of the exponential distribution is the inverted exponential distribution

proposed by Killer and Kamath (1982) which possess the inverted bathtub hazard rate and has

cumulative distribution function (CDF) expressed as

F(x,a


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