Statistical inference under mixture models

Statistical inference under mixture models / Jiahua Chen.

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 usin...

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Bibliographic Details
Main Author: Chen, Jiahua
Format: eBook
Language:English
Published: Singapore : Springer, 2023.
Series:ICSA book series in statistics.
Subjects:
Mathematical statistics.
Online Access:Click for online access
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

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