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Table of Contents
Intro
Preface
Contents
1 Introduction to Mixture Models
1.1 Mixture Model
1.2 Missing Data Structure
1.3 Identifiability
1.4 Identifiability of Some Commonly Used Mixture Models
1.4.1 Poisson Mixture Model
1.4.2 Negative Binomial Distribution
1.4.3 Finite Binomial Mixtures
1.4.4 Normal/Gaussian Mixture in Mean, in Variance, and in General
1.4.5 Finite Normal/Gaussian Mixture
1.4.6 Gamma Mixture
1.4.7 Beta Mixture
1.5 Connections Between Mixture Models
1.6 Over-Dispersion
2 Non-Parametric MLE and Its Consistency
2.1 Non-Parametric Mixture Model, Likelihood Function and the MLE
2.2 Consistency of Non-Parametric MLE
2.2.1 Distance and Compactification
2.2.2 Expand the Mixture Model Space
2.2.3 Jensen's Inequality
2.2.4 Consistency Proof of kiefer1956consistency
2.2.5 Consistency Proof of pfanzagl1988consistency
2.3 Enhanced Jensen's Inequality and Other Technicalities
2.4 Condition C20.2 and Other Technicalities
2.4.1 Summary
3 Maximum Likelihood Estimation Under Finite Mixture Models
3.1 Introduction
3.2 Generic Consistency of MLE Under Finite Mixture Models
3.3 Redner's Consistency Result
3.4 Examples
4 Estimation Under Finite Normal Mixture Models
4.1 Finite Normal Mixture with Equal Variance
4.2 Finite Normal Mixture Model with Unequal Variances
4.2.1 Unbounded Likelihood Function and Inconsistent MLE
4.2.2 Penalized Likelihood Function
4.2.3 Technical Lemmas
4.2.4 Selecting a Penalty Function
4.2.5 Consistency of the pMLE, Step I
4.2.6 Consistency of the pMLE, Step II
4.2.7 Consistency of the pMLE, Step III
4.3 Consistency When G* Has Only One Subpopulation
4.4 Consistency of the pMLE: General Order
6 Geometric Properties of Non-parametric MLE and Numerical Solutions
6.1 Geometric Properties of the Non-parametric MLE
6.2 Directional Derivative
6.3 Numerical Solutions to the Non-parametric MLE
6.4 Remarks
6.5 Algorithm Convergence
6.6 Illustration Through Poisson Mixture Model
6.6.1 Experiment with VDM
6.6.2 Experiment with VEM
6.6.3 Experiment with ISDM
7 Finite Mixture MLE and EM Algorithm
7.1 General Introduction
7.2 EM Algorithm for Finite Mixture Models
7.3 Data Examples
7.3.1 Poisson Mixture
7.3.2 Exponential Mixture
Preface
Contents
1 Introduction to Mixture Models
1.1 Mixture Model
1.2 Missing Data Structure
1.3 Identifiability
1.4 Identifiability of Some Commonly Used Mixture Models
1.4.1 Poisson Mixture Model
1.4.2 Negative Binomial Distribution
1.4.3 Finite Binomial Mixtures
1.4.4 Normal/Gaussian Mixture in Mean, in Variance, and in General
1.4.5 Finite Normal/Gaussian Mixture
1.4.6 Gamma Mixture
1.4.7 Beta Mixture
1.5 Connections Between Mixture Models
1.6 Over-Dispersion
2 Non-Parametric MLE and Its Consistency
2.1 Non-Parametric Mixture Model, Likelihood Function and the MLE
2.2 Consistency of Non-Parametric MLE
2.2.1 Distance and Compactification
2.2.2 Expand the Mixture Model Space
2.2.3 Jensen's Inequality
2.2.4 Consistency Proof of kiefer1956consistency
2.2.5 Consistency Proof of pfanzagl1988consistency
2.3 Enhanced Jensen's Inequality and Other Technicalities
2.4 Condition C20.2 and Other Technicalities
2.4.1 Summary
3 Maximum Likelihood Estimation Under Finite Mixture Models
3.1 Introduction
3.2 Generic Consistency of MLE Under Finite Mixture Models
3.3 Redner's Consistency Result
3.4 Examples
4 Estimation Under Finite Normal Mixture Models
4.1 Finite Normal Mixture with Equal Variance
4.2 Finite Normal Mixture Model with Unequal Variances
4.2.1 Unbounded Likelihood Function and Inconsistent MLE
4.2.2 Penalized Likelihood Function
4.2.3 Technical Lemmas
4.2.4 Selecting a Penalty Function
4.2.5 Consistency of the pMLE, Step I
4.2.6 Consistency of the pMLE, Step II
4.2.7 Consistency of the pMLE, Step III
4.3 Consistency When G* Has Only One Subpopulation
4.4 Consistency of the pMLE: General Order
6 Geometric Properties of Non-parametric MLE and Numerical Solutions
6.1 Geometric Properties of the Non-parametric MLE
6.2 Directional Derivative
6.3 Numerical Solutions to the Non-parametric MLE
6.4 Remarks
6.5 Algorithm Convergence
6.6 Illustration Through Poisson Mixture Model
6.6.1 Experiment with VDM
6.6.2 Experiment with VEM
6.6.3 Experiment with ISDM
7 Finite Mixture MLE and EM Algorithm
7.1 General Introduction
7.2 EM Algorithm for Finite Mixture Models
7.3 Data Examples
7.3.1 Poisson Mixture
7.3.2 Exponential Mixture