ABSTRACT
The Odd Generalized Exponential-Exponential Distribution (OGEED) could be used in various fields to model variables whose chances of success or survival decreases with time. It was also discovered that the OGEED has higher positive skewness and has been found to have performed better than some existing distributions such as the Gamma, Exponentiated Exponential, Weibull and Pareto distributions in a real life applications. The shape parameter of the Odd Generalized Exponential-Exponential Distribution using the Bayesian method of estimation and comparing the estimates with that of maximum Likelihood by assuming two non-informative prior distributions namely; Uniform and Jeffrey prior distributions. These estimates were obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF). The posterior distributions of the OGEED were derived and also the Estimates and risks were also obtained using the above mentioned priors and loss functions. Furthermore, we carried out Monte-Carlo simulation using R software to assess the performance of the two methods by making use of the Biases and MSEs of the Estimates under the Bayesian approach and Maximum likelihood method. Our result showed that Bayesian Method using Quadratic Loss Function (QLF) under both Uniform and Jeffrey priors produces the best estimates of the shape parameter compared to estimates using Maximum Likelihood method, Squared Error Loss Function (SELF) and Precautionary Loss Function (PLF) under both Uniform and Jeffrey priors irrespective of the values of the parameters and the different sample sizes. It is also discovered that the scale parameter has no effect on the estimates of the shape parameter.
TABLE OF CONTENTS
TITLE PAGE……………………………………………………………………………………………….i
DECLARATION …………………………………………………………………………………………………………………… ii
CERTIFICATION …………………………………………………………………………………………………………………. iii
DEDICATION ……………………………………………………………………………………………………………………… iv
ACKNOWLEDGEMENT ……………………………………………………………………………………………………….. v
ABSTRACT ………………………………………………………………………………………………………………………… vii
LIST OF FIGURES ………………………………………………………………………………………………………………… x
LIST OF TABLES ………………………………………………………………………………………………………………… xi
CHAPTER ONE: INTRODUCTION ………………………………………………………………………………………… 1
1.1 Background to the study ………………………………………………………………………………………………….. 1
1.2 Statement of the problem ………………………………………………………………………………………………. 2
1.3 Aim and Objectives………………………………………………………………………………………………………. 3
1.4 Significance of the Study ……………………………………………………………………………………………….. 4
CHAPTER TWO: LITERATURE REVIEW ………………………………………………………………………………. 5
2.1 BAYESIAN ESTIMATION APPROACH ……………………………………………………………………….. 5
2.2 COMPARING BAYESIAN AND MAXIMUM LIKELIHOOD ESTIMATES ……………………….. 9
CHAPTER THREE: METHODOLOGY. …………………………………………………………………………………. 11
3.1 Odd Generalized Exponential-Exponential Distribution…………………………………………………….. 11
3.2 Maximum Likelihood Estimation Method ………………………………………………………………………. 12
3.3 Posterior Distributions ………………………………………………………………………………………………… 13
3.3.1 Posterior Distribution of the Shape Parameter under the Assumption of Uniform Prior …….. 14
3.3.2 Posterior Distribution of the Shape Parameter under the Assumption of Jeffrey’s Prior …….. 16
3.4 Bayesian Estimation……………………………………………………………………………………………………. 18
3.4.1 Bayesian Estimation under Uniform Prior Using Three Loss Functions …………………………. 18
3.4.2 Bayesian Estimation under Jeffrey’s Prior Using Three Loss Functions …………………………. 26
3.5 Posterior Risks under the Priors and Loss Functions …………………………………………………………. 33
3.5.1 Posterior Risks under the Uniform Prior …………………………………………………………………… 34
3.5.2 Posterior Risks under Jeffrey’s Prior ………………………………………………………………………… 38
3.6 Simulation study ………………………………………………………………………………………………………… 42
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CHAPTER FOUR: RESULTS, ANALYSIS AND DISCUSSION ………………………………………………… 44
4.1 Simulation Studies to Compare Uniform and Jeffrey Prior …………………………………………………… 44
4.2 Discussion of Results …………………………………………………………………………………………………….. 49
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS …………………………… 51
5.1 Summary…………………………………………………………………………………………………………………… 51
5.2 Conclusion ………………………………………………………………………………………………………………… 51
5.3 Recommendations ……………………………………………………………………………………………………… 52
5.4 Contribution to Knowledge ………………………………………………………………………………………….. 52
5.5 Areas of Further Research ……………………………………………………………………………………………. 53
REFERENCES…………………………………………………………………………………………………………………….. 54
APPENDIXES……………………………………………………………………………………………56
CHAPTER ONE
INTRODUCTION
1.1 Background to the Study
In Bayesian approach, the prior information is combined with any new information that is available to form the basis for statistical inference. Statistical approaches that use prior knowledge in addition to the sample evidence to estimate the population parameters are known as Bayesian methods. The Bayesian approach seeks to optimally merge information from two sources: (1) knowledge that is known from theory or opinion formed at the beginning of the research in the form of a prior, and (2) information contained in the data in the form of likelihood functions. Basically, the prior distribution represents our initial belief, whereas the information in the data is expressed by the likelihood function. Combining prior distribution and likelihood function, we can obtain the posterior distribution. The one parameter Exponential distribution describes the time between events in a Poisson process. Its discrete analogue is the Geometric distribution. Apart from its usage in Poisson processes, it has been used extensively in the literature for life testing. The Exponential distribution is memoryless and has a constant failure rate; this latter property makes the distribution unsuitable for real life problems with bathtub failure rates (Singh et al., 2013) and inverted bathtub failure rates, hence there is need to generalize the Exponential distribution in order to increase its flexibility and capability to model some other real life problems.
There are several ways of adding one or more parameters to a distribution function which makes the resulting distribution richer and more flexible for modeling data. Some of the recent studies on the generalization of exponential distribution include the transmuted exponential distribution
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(Owoloko et al., 2015), transmuted inverse exponential distribution (Oguntunde and Adejumo, 2015), the odd generalized exponential-exponential distribution (Maiti and Pramanik, 2015) and the Weibull-Exponential distribution (Oguntunde et al., 2015). Of interest to us in this dissertation is the odd generalized exponential-exponential distribution (OGEED) which can be use in various fields to model variables whose chances of success decreases with time whereas that of failure increases as time increases. It was also discovered that the OGEED has higher positive skewness and has been found to have performed better than some existing distributions such as the Gamma, Exponentiated-Exponential, Weibull and Pareto distributions.
1.2 Statement of the Problem
Despite the nature and performance of the odd generalized exponential-exponential distribution (OGEED) compared to other distributions, there has been no study that has used Bayesian approach to estimate the shape parameter of the OGEED as at the time of this literature review. Thus, it is always of interest to study the behavior and properties of an estimator of a parameter of a new distribution.
In Bayesian inference, the performance of the estimator or predictor depends on the prior distribution and also on the loss function used. This study shall use Bayesian approach to estimate the shape parameter of the OGEED proposed by Maiti and Pramanik (2015) under the Uniform and Jeffrey Priors using the Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF). Our choice of the two priors is because we have little or no prior knowledge about the parameter of interest and so we want a prior with minimal influence on the inference to be made. The most significant thing about Bayesian method over the frequentist or classical methods is the ability of the Bayesian analysis to make use of that additional information in the form of the prior distribution. As a result of this
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incorporation, Bayesian methods do always provide stronger inferences from the same data. This is because the prior information allows the Bayesian analysis to be more responsive to the context of the data. Besides the above, Aliyu and Yahaya (2016), Preda et al. (2010) and Dey (2010) in their separate studies employed extensive Monte Carlo simulations to compare the performances of the Bayes estimates with those of Maximum Likelihood Estimates (MLEs) in Generalized Rayleigh distribution, Modified Weibull distribution and Generalized Exponential distribution respectively. It was found that the estimates under Bayesian method were more stable than the estimates under the method of MLE i.e. the non-classical or Bayes estimates have smaller risks or RMSE.
1.3 Aim and Objectives of the Study
The aim of this study is to estimate the shape parameter of Odd Generalized Exponential-Exponential Distribution (OGEED) using Bayesian approach. The specific objectives are to:
i. determine the posterior distributions under the Uniform and Jeffrey priors.
ii. obtain the Bayes estimate and Posterior risk using the Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF) under the Uniform and Jeffrey Priors.
iii. carry out a simulation study in order to find the most appropriate combination of loss functions and priors good for estimating the shape parameter of the distribution.
iv. compare Bayesian approach with the method of MLE.
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1.4 Significance of the Study
In this research, we choose two non informative priors due to the fact that we have little knowledge about the distribution of the shape parameter. Also, our interest in the shape parameter rather than scale parameter is because the shape parameter is responsible for the outstretched or degree of its skewness and kurtosis. This study will clearly estimate the shape parameter of the Odd Generalized Exponential-Exponential Distribution (OGEED) using Bayesian approach which is expected to be more stable.
This work serves as a reference material for other researchers and could be used to compare the estimate of the parameter obtained by using Bayesian method and the method of maximum likelihood estimation.
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