Parameter estimation for stochastic processes. Transl. from the Russian and ed. by B. L. S. Prakasa Rao.

*(English)*Zbl 0542.62073
Research and Exposition in Mathematics, 6. Berlin: Heldermann Verlag. VIII, 206 p. DM 56.00 (1984).

The original Russian edition of this book appeared in 1980 (see Zbl 0432.62057) and the comparison of these two editions shows that the English translation contains several new and interesting results published recently. As a whole the book is devoted to problems of parameter estimation for continuous time stochastic processes. In this review I shall indicate only some news included into the English translation.

Chapter 1 contains additional facts from mathematical statistics (Hajek inequality, analysis of locally asymptotic normality). Chapter 2 deals with essentially new results for multidimensional processes and estimation of their parameters. The original edition treats only the one- dimensional case. Detailed proofs and additional examples are given.

Chapter 3 has a little different structure. Now we find a systematic description of diffusion type processes and results about absolute continuity of measures associated with this class of processes. The author presents new and interesting statements in estimation problems in the multidimensional case, as well as for ergodic processes and processes with small diffusions.

Chapter 4 deals with a more general class of processes, namely the class of point processes. Referring the reader to chapter 17 and chapter 18 of the English edition of the book by A. N. Shiryayev and R. S. Liptser, Statistics of random processes, II. (1978; Zbl 0369.60001), the author presents essentially new results and examples concerning statistics of point processes. Chapter 5 again contains new and nice results about the LAN families and asymptotic properties of statistical estimators.

Let us note that in the original edition references are given after each chapter. The English translation includes a unified and extended reference list at the end of the book which is more convenient for the readers. The book is well translated and edited. Thus, the book under review, being a translation of a completely revised and extended version of the original edition, presents an important branch of modern stochastics and its applications. The content of the book, the organization of the material and the style of presentation make it appropriate for stochasticians doing research in this field, for various applications of stochastics, and finally, for special courses addressed to graduate university students.

Chapter 1 contains additional facts from mathematical statistics (Hajek inequality, analysis of locally asymptotic normality). Chapter 2 deals with essentially new results for multidimensional processes and estimation of their parameters. The original edition treats only the one- dimensional case. Detailed proofs and additional examples are given.

Chapter 3 has a little different structure. Now we find a systematic description of diffusion type processes and results about absolute continuity of measures associated with this class of processes. The author presents new and interesting statements in estimation problems in the multidimensional case, as well as for ergodic processes and processes with small diffusions.

Chapter 4 deals with a more general class of processes, namely the class of point processes. Referring the reader to chapter 17 and chapter 18 of the English edition of the book by A. N. Shiryayev and R. S. Liptser, Statistics of random processes, II. (1978; Zbl 0369.60001), the author presents essentially new results and examples concerning statistics of point processes. Chapter 5 again contains new and nice results about the LAN families and asymptotic properties of statistical estimators.

Let us note that in the original edition references are given after each chapter. The English translation includes a unified and extended reference list at the end of the book which is more convenient for the readers. The book is well translated and edited. Thus, the book under review, being a translation of a completely revised and extended version of the original edition, presents an important branch of modern stochastics and its applications. The content of the book, the organization of the material and the style of presentation make it appropriate for stochasticians doing research in this field, for various applications of stochastics, and finally, for special courses addressed to graduate university students.

Reviewer: J.M.Stoyanov

##### MSC:

62M05 | Markov processes: estimation; hidden Markov models |

62M09 | Non-Markovian processes: estimation |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62Mxx | Inference from stochastic processes |