Multivariate Exponential Families: A Concise Guide to Statistical Inference (SpringerBriefs in Statistics)

✍ Scribed by Stefan Bedbur

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 147
• Category: Library

✦ Synopsis

This book provides a concise introduction to exponential families. Parametric families of probability distributions and their properties are extensively studied in the literature on statistical modeling and inference. Exponential families of distributions comprise density functions of a particular form, which enables general assertions and leads to nice features. With a focus on parameter estimation and hypotheses testing, the text introduces the reader to distributional and statistical properties of multivariate and multiparameter exponential families along with a variety of detailed examples. The material is widely self-contained and written in a mathematical setting. It may serve both as a concise, mathematically rigorous course on exponential families in a systematic structure and as an introduction to Mathematical Statistics restricted to the use of exponential families.

✦ Table of Contents

Preface
Contents
Symbols
1 Introduction: Aims and Outline
2 Parametrizations and Basic Properties
2.1 Definition and Examples
2.2 Order and Minimal Representation
2.3 Structural Properties and Natural Parametrization
2.4 Analytical Properties and Mean Value Parametrization
3 Distributional and Statistical Properties
3.1 Generating Functions
3.2 Marginal and Conditional Distributions
3.3 Product Measures
3.4 Sufficiency and Completeness
3.5 Score Statistic and Information Matrix
3.6 Divergence and Distance Measures
4 Parameter Estimation
4.1 Maximum Likelihood Estimation
4.2 Constrained Maximum Likelihood Estimation
4.3 Efficient Estimation
4.4 Asymptotic Properties
5 Hypotheses Testing
5.1 One-Sided Test Problems
5.2 Two-Sided Test Problems
5.3 Testing for a Selected Parameter
5.4 Testing for Several Parameters
6 Exemplary Multivariate Applications
6.1 Negative Multinomial Distribution
6.2 Dirichlet Distribution
6.3 Generalized Order Statistics
References
Index