Statistical Inference Under Mixture Models (ICSA Book Series in Statistics)

✍ Scribed by Jiahua Chen

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 330
• Category: Library

✦ Synopsis

This book puts its weight on theoretical issues related to finite mixture models. It shows that a good applicant, is an applicant who understands the issues behind each statistical method. This book is intended for applicants whose interests include some understanding of the procedures they are using, while they do not have to read the technical derivations.

At the same time, many researchers find most theories and techniques necessary for the development of various statistical methods, without chasing after one set of research papers, after another. Even though the book emphasizes the theory, it provides accessible numerical tools for data analysis. Readers with strength in developing statistical software, may find it useful.

✦ Table of Contents

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
4.5 Consistency of the pMLE Under Multivariate Finite Normal Mixture Models
4.5.1 Some Remarks
4.6 Asymptotic Normality of the Mixing Distribution Estimation
5 Consistent Estimation Under Finite Gamma Mixture
5.1 Upper Bounds on f(x; r, θ)
5.2 Consistency for Penalized MLE
5.2.1 Likelihood Function on G1
5.2.2 Likelihood Function on G2
5.2.3 Consistency on G3 and Consistency for All
5.3 Consistency of the Constrained MLE
5.4 Example
5.4.1 Some Simulation Results
5.5 Consistency of the MLE When Either Shape or Scale Parameter Is Structural
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
7.4 Convergence of the EM Algorithm under Finite Mixture Model
7.4.1 Global Convergence Theorem
7.4.2 Convergence of EM Algorithm Under Finite Mixture
7.5 Discussions
8 Rate of Convergence
8.1 Example
8.2 Best Possible Rate of Convergence
8.3 Strong Identifiability and Minimum Distance Estimator
8.4 Other Results on Best Possible Rates
8.5 Strongly Identifiable Distribution Families
8.6 Impact of the Best Minimax Rate Is Not n-1/4
9 Test of Homogeneity
9.1 Test for Homogeneity
9.2 Hartigan's Example
9.3 Binomial Mixture Example
9.4 C(α) Test
9.4.1 The Generic C(α) Test
9.4.2 C(α) Test for Homogeneity
9.4.3 C(α) Statistic Under NEF-QVF
9.4.4 Expressions of the C(α) Statistics forNEF-VEF Mixtures
10 Likelihood Ratio Test for Homogeneity
10.1 Likelihood Ratio Test for Homogeneity: One Parameter Case
10.2 Examples
10.3 The Proof of Theorem 10.1
10.3.1 An Expansion of Under the Null Model
10.3.2 Expansion of R1n: Preparations
10.3.3 Expanding R1n
10.4 Homogeneity Under Normal Mixture Model
10.4.1 The Result for a Single Mean Parameter
10.5 Two-Mean Parameter Mixtures: Tests for Homogeneity
10.5.1 Large Sample Behavior of the MLE's
10.5.2 Analysis of Rn(ε;I)
10.5.3 Analysis of Rn(ε;II)
10.5.4 Asymptotic Distribution of the LRT
11 Modified Likelihood Ratio Test
11.1 Test of Homogeneity with Binomial Observations
11.2 Modified Likelihood Ratio Test for Homogeneity with Multinomial Observations
11.3 Test for Homogeneity for General Subpopulation Distribution
11.4 Test for Homogeneity in the Presence of a Structural Parameter
12 Modified Likelihood Ratio Test for Higher Order
12.1 Test for Higher Order
12.2 A Modified Likelihood Ratio Test for m=2
12.2.1 Technical Outlines
12.2.2 Regularity Conditions and Rigorous Proofs
13 em-Test for Homogeneity
13.1 Limitations of the Modified Likelihood Ratio Test
13.2 em-Test for Homogeneity
13.3 The Asymptotic Properties
13.4 Precision-Enhancing Measures
14 em-Test for Higher Order
14.1 Introduction
14.2 The em-Test Statistic
14.3 The Limiting Distribution
14.3.1 Outline of the Proof
14.3.2 Conditions Underlying the Limiting Distribution of the em-Test
14.4 Tuning Parameters
14.5 Data Examples
15 em-Test for Univariate Finite Gaussian Mixture Models
15.1 Introduction
15.2 The Construction of the em-Test
15.3 Asymptotic Results
15.4 Choice of Various Influential Factors
15.5 Data Example
16 Order Selection of the Finite Mixture Models
16.1 Order Selection
16.2 Selection via Classical Information Criteria
16.3 Variations of the Information Criterion
16.4 Shrinkage-Based Approach
17 A Few Key Probability Theory Results
17.1 Introduction
17.2 Borel-Cantelli Lemma
17.3 Random Variables and Stochastic Processes
17.4 Uniform Strong Law of Large Numbers
References
Index