Asymptotic Statistics: With a View to Stochastic Processes

by Reinhard Höpfner

Overview

This textbook is devoted to the general asymptotic theory of statistical experiments. Local asymptotics for statistical models in the sense of local asymptotic (mixed) normality or local asymptotic quadraticity make up the core of the book. Numerous examples deal with classical independent and identically distributed models and with stochastic processes.

Product Details

ISBN-13:	9783110367782
Publisher:	De Gruyter
Publication date:	05/26/2014
Series:	De Gruyter Textbook
Sold by:	Barnes & Noble
Format:	eBook
Pages:	286
File size:	11 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Reinhard Höpfner, Johannes Gutenberg University Mainz, Germany.

Table of Contents

Preface vii

1 Score and Information 1

1.1 Score, Information, Information Bounds 2

1.2 Estimator Sequences, Asymptotics of Information Bounds 15

1.3 Heuristics on Maximum Likelihood Estimator Sequences 23

1.4 Consistency of ML Estimators via Hellinger Distances 30

2 Minimum Distance Estimators 42

2.1 Stochastic Processes with Paths in L^P (T, T, μ) 43

2.2 Minimum Distance Estimator Sequences 55

2.3 Some Comments on Gaussian Processes 68

2.4 Asymptotic Normality for Minimum Distance Estimator Sequences 75

3 Contiguity 85

3.1 Le Cam's First and Third Lemma 86

3.2 Proofs for Section 3.1 and some Variants 92

4 L²-differentiable Statistical Models 108

4.1 L^r-differentiate Statistical Models 109

4.2 Le Cam's Second Lemma for i.i.d. Observations 119

5 Gaussian Shift Models 127

5.1 Gaussian Shift Experiments 127

5.2 *Brownian Motion with Unknown Drift as a Gaussian Shift Experiment 141

6 Quadratic Experiments and Mixed Normal Experiments 148

6.1 Quadratic and Mixed Normal Experiments 148

6.2 *Likehhood Ratio Processes in Diffusion Models 160

6.3 *Time Changes for Brownian Motion with Unknown Drift 168

7 Local Asymptotics of Type LAN, LAMN, LAQ 178

7.1 Local Asymptotics of Type LAN. LAMN, LAQ 179

7.2 Asymptotic optimality of estimators in the LAN or LAMN setting 191

7.3 Le Cam's One-step Modification of Estimators 200

7.4 The Case of i.i.d. Observations 206

8 *Some Stochastic Process Examples for Local Asymptotics of Type LAN, LAMN and LAQ 212

8.1 *Ornstein-Uhlenbeck Process with Unknown Parameter Observed over a Long Time Interval 213

8.2 *A Null Recurrent Diffusion Model 227

8.3 *Some Further Remarks 240

*Appendix 243

9.1 *Convergence of Martingales 244

9.2 *Harris Recurrent Markov Processes 247

9.3 *Checking the Harris Condition 253

9.4 * One-dimensional Diffusions 258

Bibliography 267

Index 275