Empirical Bayes Estimators of Positive Parameters in Hierarchical Models under Stein's Loss Function.

Format:

Book

Author/Creator:

Zhang, Yingying.

Series:

Current Natural Sciences Series

Language:

English

Physical Description:

1 online resource (358 pages)

Edition:

1st ed.

Place of Publication:

Les Ulis : EDP Sciences, 2025.

Summary:

This book presents in-depth research on positive parameters of hierarchical models under Stein's loss function and proposes a novel empirical Bayesian estimation method.

Contents:

Intro

Empirical Bayes Estimators of Positive Parameters in Hierarchical Models under Stein's Loss Function

Preface

Contents

List of Figures

List of Tables

List of Abbreviations

Introduction

Empirical Bayes Method

The Gamma and Inverse Gamma Distributions

Hierarchical Models with Positive Parameters

Estimating the Hyperparameters

Stein's Loss Function

The Bayes Estimators and the PESLs

Theoretical Comparisons of the Bayes Estimators and the PESLs of Three Methods

Simulation Techniques

Consistencies of the Moment Estimators and the MLEs

Goodness-of-Fit of the Model

Numerical Comparisons of the Bayes Estimators and the PESLs of Three Methods

R Codes

The Empirical Bayes Estimators of the Rate Parameter of the Inverse Gamma Distribution with a Conjugate Inverse Gamma Prior under Stein's Loss Function

Theoretical Results

The Empirical Bayes Estimators of θn+1

Simulations

Two Inequalities of the Bayes Estimators and the PESLs

Goodness-of-Fit of the Model: KS Test

Marginal Densities for Various Hyperparameters

Conclusions and Discussions

The Empirical Bayes Estimators of the Rate Parameter of the Gamma Distribution with a Conjugate Gamma Prior under Stein's Loss Function

Consistencies of the Moment Estimators and the MLEs.

Goodness-of-Fit of the Model: KS Test

The Empirical Bayes Estimators of the Mean Parameter of the Exponential Distribution with a Conjugate Inverse Gamma Prior under Stein's Loss Function

A Real Data Example

The Empirical Bayes Estimators of the Variance Parameter of the Normal Distribution with a Conjugate Inverse Gamma Prior under Stein's Loss Function

The Empirical Bayes Estimators of the Variance Parameter of the Normal Distribution with a Normal-Inverse-Gamma Prior under Stein's Loss Function

The Bayes Estimators and the PESLs.

The Empirical Bayes Estimators of θn+1

The Empirical Bayes Estimators of the Parameter of the Uniform Distribution with an Inverse Gamma Prior under Stein's Loss Function

The Empirical Bayes Estimators of the Parameter of the Poisson Distribution with a Conjugate Gamma Prior under Stein's Loss Function

Goodness-of-Fit of the Model: Chi-Square Test

Marginal pmfs for Various Hyperparameters

Several Common Loss Functions

Two Loss Functions on Θ = (-∞, ∞)

Squared Error Loss Function

Weighted Squared Error Loss Function

Power-Power Loss Function

Two Loss Functions on Θ = (0, 1)

Power-Log Loss Function

Zhang's Loss Function

Three Strings of Inequalities among Six Bayes Estimators

Other Loss Functions

LINEX Loss Function

Absolute Error Loss Function

Weighted Absolute Error Loss Function.

Power Loss Function

Weighted Power Loss Function

Log-1 Loss Function

Log-2 Loss Function

Generalized Log Loss Function

Generalized Stein's Loss Function

Generalized Power-Power Loss Function

Summary of the Loss Functions

Summaries and Discussions

Appendix A

A.1 IG-IG: The Proof of Theorem 2.1

A.2 IG-IG: The Proof of Theorem 2.2

A.3 IG-IG: The Proof of Theorem 2.3

A.4 IG-IG: The Simulation Design of Subsection 2.3.2

A.5 G-G: The Proof of Theorem 3.1

A.6 G-G: The Calculations of E(logθn+1Ixn+1) and the Two PESLs

A.7 G-G: The Proof of Theorem 3.2

A.8 G-G: The Proof of Theorem 3.3

A.9 Exp-IG: The Proof of Theorem 4.1

A.10 Exp-IG: The Proof of Theorem 4.2

A.11 Exp-IG: The Proof of Theorem 4.3

A.12 N-IG: The Proof of Theorem 5.1

A.13 N-IG: The Proof of Lemma 5.1

A.14 N-IG: The Proof of Lemma 5.2

A.15 N-IG: The Proof of Lemma 5.3

A.16 N-IG: The Proof of Lemma 5.4

A.17 N-IG: The Proof of Theorem 5.2

A.18 N-IG: The Proof of Theorem 5.3

A.19 N-NIG: The Proof of Theorem 6.1

A.20 N-NIG: The Proof of Theorem 6.2

A.21 N-NIG: The Proof of Theorem 6.3

A.22 U-IG: The Proof of Theorem 7.1

A.23 U-IG: Some Key Notations and Derivatives

A.24 U-IG: Tedious and Complicated Calculations of E1, E2, and E3

A.25 U-IG: The Proof of Theorem 7.2

A.26 U-IG: The Proof of Theorem 7.3

A.27 U-IG: The Analytical Calculations of Int

A.28 P-G: The Proof of Theorem 8.1

A.29 P-G: The Proof of Theorem 8.2

A.30 P-G: The Proof of Theorem 8.3

Appendix B

B.1 Univariate Continuous Distributions

B.2 Univariate Discrete Distributions

References.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

2-7598-3912-5

9782759839124

OCLC:

1564936709