Asymptotic Statistics: With a View to Stochastic Processes

✍ Scribed by Reinhard Höpfner

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 286
• Category: Library

✦ Synopsis

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.

The book can be read in different ways, according to possibly different mathematical preferences of the reader. One reader may focus on the statistical theory, and thus on the chapters about Gaussian shift models, mixed normal and quadratic models, and on local asymptotics where the limit model is a Gaussian shift or a mixed normal or a quadratic experiment (LAN, LAMN, LAQ). Another reader may prefer an introduction to stochastic process models where given statistical results apply, and thus concentrate on subsections or chapters on likelihood ratio processes and some diffusion type models where LAN, LAMN or LAQ occurs. Finally, readers might put together both aspects.

The book is suitable for graduate students starting to work in statistics of stochastic processes, as well as for researchers interested in a precise introduction to this area.

• Introduction for readers who start to work in statistics of stochastic processes
• Presentation of local asymptotics in statistics
• Presentation of stochastic process models
• Includes numerous examples and exercises
• Also attractive for probabilists

✦ Table of Contents

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