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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.
xiii
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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