##
**Comparison methods for stochastic models and risks.**
*(English)*
Zbl 0999.60002

Wiley Series in Probability and Statistics. Chichester: Wiley. xii, 330 p. (2002).

It is not an easy task to write a book on stochastic orders and their applications because the theory of stochastic orders is still in intensive development and has many important fields of applications. The authors have written a very timely and methodologically oriented book on stochastic orders and their applications which can be used by a wide spectrum of readers and users in applied probability and statistics, reliability, queueing theory, economics and finance, and actuarial science.

The book contains eight chapters. Contents of chapters are: 1. Univariate stochastic orders (including several common used stochastic orders, interrelationships among them, bivariate characterizations and aging notions of life distributions); 2. Theory of integral stochastic orders (from a very general viewpoint, including maximal and small generators, preservation properties of orders from their generators, and Strassen theorem); 3. Multivariate stochastic orders (including the possible generalizations of the usual stochastic order, multivariate variability orders like convex order and its generalizations, an extensive treatment of dependence orders and characterizations of dependence structures of random vectors, and stochastic ordering of multivariate normal distributions); 4. Stochastic models, comparison and monotonicity (including general principles for deriving comparison and monotonicity results for stochastic models); 5. Monotonicity and comparability of stochastic processes (including a thorough survey on comparison properties of Markov processes (especially, Markov chains), and some comparison results for non-Markov processes and point processes); 6. Monotonicity properties and bounds for queueing systems (including the classical applications of stochastic orders in queueing systems, and some examples for the application of dependence orders); 7. Applications to various stochastic models (including some applications of stochastic orders in reliability theory, statistical physics, scheduling and others); 8. Comparing risks (including applications in economics and actuarial science).

The book contains eight chapters. Contents of chapters are: 1. Univariate stochastic orders (including several common used stochastic orders, interrelationships among them, bivariate characterizations and aging notions of life distributions); 2. Theory of integral stochastic orders (from a very general viewpoint, including maximal and small generators, preservation properties of orders from their generators, and Strassen theorem); 3. Multivariate stochastic orders (including the possible generalizations of the usual stochastic order, multivariate variability orders like convex order and its generalizations, an extensive treatment of dependence orders and characterizations of dependence structures of random vectors, and stochastic ordering of multivariate normal distributions); 4. Stochastic models, comparison and monotonicity (including general principles for deriving comparison and monotonicity results for stochastic models); 5. Monotonicity and comparability of stochastic processes (including a thorough survey on comparison properties of Markov processes (especially, Markov chains), and some comparison results for non-Markov processes and point processes); 6. Monotonicity properties and bounds for queueing systems (including the classical applications of stochastic orders in queueing systems, and some examples for the application of dependence orders); 7. Applications to various stochastic models (including some applications of stochastic orders in reliability theory, statistical physics, scheduling and others); 8. Comparing risks (including applications in economics and actuarial science).

Reviewer: Moshe Shaked (Tucson)

### MSC:

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

60E15 | Inequalities; stochastic orderings |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90B25 | Reliability, availability, maintenance, inspection in operations research |