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Title of Thesis

Bayesian Analysis Of Mixture Distributions

Author(s)

Muhammad Saleem

Institute/University/Department Details
Department Of Statistics / Quaid-i-azam University, Islamabad
Session
2010
Subject
Statistics
Number of Pages
142
Keywords (Extracted from title, table of contents and abstract of thesis)
Classical, Relevant, Mixture, Bayes, Distributions, Bayesian, Properties, Comparison, Subclass, Real, Estimators, Analysis

Abstract
In this thesis, we consider type-I mixtures of the members of a subclass of one parameter exponential family.This subclass includes Exponential, Rayleigh, Pareto, a Burr type XII and Power function distributions. Except the Exponential, mixtures of distributions of this subclass get either no or least attention in literature so far.
The elegant closed form expressions for the Bayes estimators of the parameters of each of these mixtures are presented along with their variances assuming uninformative and informative priors.The proposed informative Bayes estimators emerge advantageous in terms of their least standard errors.An extensive simulation study is conducted for each of these mixtures to highlight the properties and comparison of the proposed Bayes estimators in terms
of sample sizes, censoring rates, mixing proportions and different combinations of the parameters of the component densities. A type-IV sample consisting of ordinary type-I, right censored observations is considered. Bayesian analysis of the real life mixture data sets is conducted as an application of each mixture and some interesting observations and comparisons have been observed.
The systems of non-linear equations to evaluate the classical maximum likelihood estimates, the components of the information matrices, complete sample expressions, the posterior predictive distributions and the equations for the evaluation of the Bayesian predictive intervals are derived for each of these mixtures as relevant algebra. The predictive intervals are evaluated in case of the Rayleigh mixture only for a number of combinations of the hyperparameters to look for a trend among the hyperparameters that may lead towards an efficient estimation.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

 
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2

1

INTRODUCTION AND REVIEW OF LITERATURE

 

1
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3 2 PROPERTIES AND COPARISON OF THE BAYES ESTIMATES OF THE EXPONENTIAL FINITE MIXTURE PARAMETERS

2.1 Introduction
2.2 The Exponential Finite Mixture Model
2.3 The Maximum Likelihood Estimates for Censored Data
2.4 The Posterior Distributions assuming the Conjugate Prior
2.5 Bayes Estimators assuming Uninformative Priors
2.6 The Complete Sample Expressions
2.7 A Simulation Study
2.8 A Real Life Example
2.9 Conclusion

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4 3 PROPERTIES AND COMPARISON OF THE BAYES ESTIMATES OF THE RAYLEIGH FINITE MIXTURE PARAMETERS

3.1 Introduction
3.2 The Rayleigh Mixture Model
3.3 The Maximum Likelihood Estimates for Censored Data
3.4 The Posterior Distributions assuming the Conjugate Prior
3.5 The Posterior Distributions assuming the Uninformative Priors
3.6 The Complete Sample Expressions
3.7 A Simulation Study
3.8 A Real Life Example
3.9 Conclusion

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5 4 THE PRIOR SELECTION FOR THE MIXTURE OF THE RAYLEIGH DISTRIBUTION USING PREDICTIVE INTERVALS

4.1 Introduction
4.2 The Rayleigh Mixture Model
4.3 Sampling
4.4 The Posterior Distribution assuming the Inverted Chi Prior
4.5 The Posterior Distribution assuming the Inverted Rayleigh Prior
4.6 The Posterior Distribution assuming the Square Root Inverted Gamma Prior
4.7 An Example Based on Simulated Data 53
4.8 Conclusion

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6 5 ESTIMATION AND APPLICATION OF THE PARETO MIXTURE

5.1 Introduction
5.2 The Maximum Likelihood Estimates for Censored Data
5.3 Bayes Estimators assuming the Conjugate Prior
5.4 The Complete Sample Expressions
5.5 A Simulation Study
5.6 A Real Life Example
5.7 Conclusion

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7 6 PROPERTIES AND COMPARISON OF THE BAYES ESTIMAES OF THE BURR Finite MIXTURE PARAMETERS

6.1 Introduction
6.2 A Burr Finite Mixture Model
6.3 The Maximum Likelihood Estimates for Censored Data
6.4 The Posterior Distributions assuming the Uninformative Priors
6.5 The Posterior Distributions assuming the Congugate Prior
6.6 The Complete Sample Expressions
6.7 A Simulation Study
6.8 A Real Life Example
6.9 Conclusion

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8 7 PROPERTIES AND COMPARISON OF THE POWER FUNCTION MIXTURE PARAMETERS

7.1 Introduction
7.2 A Power Function Mixture Model
7.3 The Maximum Likelihood Estimates for Censored Data
7.4 The Posterior Distributions assuming the Uninformative Priors
7.5 The Posterior Distributions assuming the Conjugate Prior
7.6 The Complete Sample Expressions
7.7 A Simulation Study
7.8 A Real Life Example
7.9 Conclusion

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9 8 CONCLUSION AND RECOMMENDATIONS

 

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10 9 REFERENCES

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