Stochastic orders and their applications.

*(English)*Zbl 0806.62009
Probability and Mathematical Statistics. Boston, MA: Academic Press. xvi, 545 p. (1994).

This book gives in the first part (6 chapters) an informal exposition of stochastic ordering results by the authors. The second part (10 chapters) is a collection of papers of several authors concerned with applications of stochastic orders to various fields like reliability, networks, operations research, comparing risks, epidemics and to statistics.

The first part gives a well written collection of results on stochastic ordering which are of interest to users of stochastic orderings in several application areas. In this sense the authors present results, sometimes with proofs, which are motivated and of interest for the applications. This part gives the applied scientist a collection of tools available w.r.t. the most common types of stochastic orders. It gives a lot of valuable and useful references in a practical sense, but not always in a historical sense. E.g. concerning the stochastic ordering (Theorem 4.B.1) it is said that this result can be found in T. Kamae, U. Krengel and G. L. O’Brien [Ann. Probab. 5, 899-912 (1977; Zbl 0371.60013)], while the original paper of V. Strassen [Ann. Math. Stat. 36, 423-439 (1965; Zbl 0135.187)] is not mentioned in this context. Some of the available a.s. representation results are also stated for each order separately (some are missing) while under a more methodologically oriented concept it would perhaps have been more natural, to state a unique general version of this result valid for all types of ordering.

The second part gives 10 lectures on the application of stochastic orders to various fields where the reader gets typically a good and not too technical introduction to the typical problems. Some of the authors of this second part formulate very explicitly the problems and special considerations in their field and this unifying character makes the book particular interesting from this point of view.

Altogether, this is a useful publication and addition to the few existing monographs on this field from a general applied view-point. It is well written, clearly organized and should be very helpful as a toolbox for the applied researcher.

The first part gives a well written collection of results on stochastic ordering which are of interest to users of stochastic orderings in several application areas. In this sense the authors present results, sometimes with proofs, which are motivated and of interest for the applications. This part gives the applied scientist a collection of tools available w.r.t. the most common types of stochastic orders. It gives a lot of valuable and useful references in a practical sense, but not always in a historical sense. E.g. concerning the stochastic ordering (Theorem 4.B.1) it is said that this result can be found in T. Kamae, U. Krengel and G. L. O’Brien [Ann. Probab. 5, 899-912 (1977; Zbl 0371.60013)], while the original paper of V. Strassen [Ann. Math. Stat. 36, 423-439 (1965; Zbl 0135.187)] is not mentioned in this context. Some of the available a.s. representation results are also stated for each order separately (some are missing) while under a more methodologically oriented concept it would perhaps have been more natural, to state a unique general version of this result valid for all types of ordering.

The second part gives 10 lectures on the application of stochastic orders to various fields where the reader gets typically a good and not too technical introduction to the typical problems. Some of the authors of this second part formulate very explicitly the problems and special considerations in their field and this unifying character makes the book particular interesting from this point of view.

Altogether, this is a useful publication and addition to the few existing monographs on this field from a general applied view-point. It is well written, clearly organized and should be very helpful as a toolbox for the applied researcher.

Reviewer: L.Rüschendorf (Freiburg)

##### MSC:

62E10 | Characterization and structure theory of statistical distributions |

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

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60E05 | Probability distributions: general theory |

62N05 | Reliability and life testing |