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**Convex statistical distances.**
*(English)*
Zbl 0656.62004

Teubner-Texte zur Mathematik, Bd. 95. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. 224 p.; DM 23.50 (1987).

In the authors’ book one finds for the first time a systematic development of properties of divergence type distances, a notion introduced by I. Csiszar [see Publ. Math. Inst. Hung. Acad. Sci., Ser. A 8, 85-108 (1963; Zbl 0124.087)]. There are ten sections of altogether 220 pages. The first two of them describe some fundamental results, the monotonicity theorem and its relation to sufficiency, the approximation theorem - a generalization of Pinsker’s theorem on divergences -, the lower semicontinuity theorem and the convexity theorem, which are all consequences of the convexity properties of divergences. Also, some of the basic divergences are introduced and discussed.

The next sections are concerned with the distance of stochastic processes with independent increments and with applications to the distribution of their likelihoods. In particular, there are formulas for the Reiny distance of Gaussian processes, Poisson processes and for general processes with independent increments in terms of their local characteristics.

Section 6 gives a short introduction to stochastic analysis and introduces Hellinger processes and -exponentials. This notion is applied to derive bounds for the Hellinger- and total variation distance of probabilities on filtered spaces.

Finally, there are applications to contiguity and entire separation, generalized Cramér-Rao inequalities and minimum distance type point estimation. The last applications are partially based on explicit results for projections w.r.t. divergences.

The book is very well written and contains many new results. The statistical applications will surely increase in the future. For work in this field this book is an excellent basis.

The next sections are concerned with the distance of stochastic processes with independent increments and with applications to the distribution of their likelihoods. In particular, there are formulas for the Reiny distance of Gaussian processes, Poisson processes and for general processes with independent increments in terms of their local characteristics.

Section 6 gives a short introduction to stochastic analysis and introduces Hellinger processes and -exponentials. This notion is applied to derive bounds for the Hellinger- and total variation distance of probabilities on filtered spaces.

Finally, there are applications to contiguity and entire separation, generalized Cramér-Rao inequalities and minimum distance type point estimation. The last applications are partially based on explicit results for projections w.r.t. divergences.

The book is very well written and contains many new results. The statistical applications will surely increase in the future. For work in this field this book is an excellent basis.

Reviewer: L.Rüschendorf

### MSC:

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

62E99 | Statistical distribution theory |

60E99 | Distribution theory |