Asymptotic Statistics (eBook)

With a View to Stochastic Processes

Reinhard Höpfner (Autor)

eBook Download: PDF | EPUB

2014 | 1. Auflage
286 Seiten
Walter de Gruyter GmbH & Co.KG (Verlag)
978-3-11-025028-2 (ISBN)

Infos

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.

Reinhard Höpfner, Johannes Gutenberg University Mainz, Germany.

Inhalt

Preface 7
1 Score and Information 11
1.1 Score, Information, Information Bounds 12
1.2 Estimator Sequences, Asymptotics of Information Bounds 25
1.3 Heuristics on Maximum Likelihood Estimator Sequences 33
1.4 Consistency of ML Estimators via Hellinger Distances 40
2 Minimum Distance Estimators 52
2.1 Stochastic Processes with Paths in Lp(T, t, µ) 53
2.2 Minimum Distance Estimator Sequences 65
2.3 Some Comments on Gaussian Processes 78
2.4 Asymptotic Normality for Minimum Distance Estimator Sequences 85
3 Contiguity 95
3.1 Le Cam’s First and Third Lemma 96
3.2 Proofs for Section 3.1 and some Variants 102
4 L2-differentiable Statistical Models 118
4.1 Lr -differentiable Statistical Models 119
4.2 Le Cam’s Second Lemma for i.i.d. Observations 129
5 Gaussian Shift Models 137
5.1 Gaussian Shift Experiments 137
5.2 Brownian Motion with Unknown Drift as a Gaussian Shift Experiment 151
6 Quadratic Experiments and Mixed Normal Experiments 158
6.1 Quadratic and Mixed Normal Experiments 158
6.2 Likelihood Ratio Processes in Diffusion Models 170
6.3 Time Changes for Brownian Motion with Unknown Drift 178
7 Local Asymptotics of Type LAN, LAMN, LAQ 188
7.1 Local Asymptotics of Type LAN, LAMN, LAQ 189
7.2 Asymptotic optimality of estimators in the LAN or LAMN setting 201
7.3 Le Cam’s One-step Modification of Estimators 210
7.4 The Case of i.i.d. Observations 216
8 Some Stochastic Process Examples for Local Asymptotics of Type LAN, LAMN and LAQ 222
8.1 Ornstein–Uhlenbeck Process with Unknown Parameter Observed over a Long Time Interval 223
8.2 A Null Recurrent DiffusionModel 237
8.3 Some Further Remarks 250
Appendix 253
9.1 Convergence of Martingales 254
9.2 Harris RecurrentMarkov Processes 257
9.3 Checking the Harris Condition 263
9.4 One-dimensional Diffusions 268
Bibliography 277
Index 285

Produkt-Details

• Erscheint lt. Verlag: 26.5.2014
• Sprache: englisch
• Themenwelt: Geisteswissenschaften
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Sozialwissenschaften ► Pädagogik
• Themenwelt: Technik
• ISBN-10: 3-11-025028-4 / 3110250284
• ISBN-13: 978-3-11-025028-2 / 9783110250282