**Translations of Mathematical Monographs**

2001;
216 pp;
Hardcover

MSC: Primary 62;
Secondary 60

**Print ISBN: 978-0-8218-1183-2
Product Code: MMONO/196**

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

**Electronic ISBN: 978-1-4704-4622-2
Product Code: MMONO/196.E**

List Price: $97.00

AMS Member Price: $77.60

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# Asymptotic Statistical Methods for Stochastic Processes

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*Yu. N. Lin′kov*

The asymptotic properties of the likelihood ratio play an important part in
solving problems in statistics for various schemes of observations. In this
book, the author describes the asymptotic methods for parameter estimation and
hypothesis testing based on asymptotic properties of the likelihood ratios in
the case where an observed stochastic process is a semimartingale.

Chapter 1 gives the general basic notions and results of the theory under
consideration. Chapters 2 and 3 are devoted to the problem of distinguishing
between two simple statistical hypotheses. In Chapter 2, certain types of
asymptotic distinguishability between families of hypotheses are
introduced. The types are characterized in terms of likelihood ratio, Hellinger
integral of order \(\epsilon\), Kakutani-Hellinger distance, and the distance in
variation between hypothetical measures, etc. The results in Chapter 2 are used
in Chapter 3 in statistical experiments generated by observations of
semimartingales. Chapter 4 applies the general limit theorems on asymptotic
properties of maximum likelihood and Bayes estimates obtained by Ibragimov and
Has'minskii for observations of an arbitrary nature to observations of
semimartingales. In Chapter 5, an unknown parameter is assumed to be random,
and under this condition, certain information-theoretic problems of estimation
of parameters are considered.

This English edition includes an extensive list of references and revised
bibliographical notes.

#### Readership

Graduate students and research mathematicians interested in statistics; engineers.

#### Table of Contents

# Table of Contents

## Asymptotic Statistical Methods for Stochastic Processes

- Cover Cover11
- Title page v6
- Contents vii8
- Preface ix10
- Basic notation xiii14
- Local densities of measures and limit theorems for stochastic processes 118
- Asymptotic distinguishing between simple hypotheses in the scheme of general statistical experiments 3350
- Asymptotic behavior of the likelihood ratio in problems of distinguishing between simple hypotheses for semimartingales 7996
- Asymptotic estimation of parameters 137154
- Asymptotic information-theoretic problems in parameter estimation 161178
- Bibliographical notes 197214
- References 203220
- Index 213230
- Back Cover Back Cover1234