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Unimodality of probability measures. (English) Zbl 0876.60001
Mathematics and its Applications (Dordrecht). 382. Dordrecht: Kluwer Academic Publishers. xiv, 251 p. (1997).
As the title claims, this book is devoted to unimodal distributions in a wide sense. It is very thoroughly written, presenting a rather complete treatment of the topic unimodality and various generalizations including to Hilbert spaces. As the authors point out, the monograph is written for understanding the concept unimodality, not as an encyclopedia. Nevertheless the presented topics are rather detailed and exhaustive, an up to date reference for unimodality. The various ideas and concepts are unified under a single point of view via Choquet type representations. For sure the book is worthwhile to read and will become a standard reference book for all those working in the area unimodality. The success will depend on how many more important examples will be added in the future. The basis is laid and the given examples are promising.
The mathematical starting point is the observation, that any probability measure on $$\mathbb{R}^+$$ having decreasing density is an integral over uniform distributions on $$[0,.]$$. In probabilistic language, it is representable as $$UZ$$ by independent r.v.s. $$U$$, $$Z$$ where $$U$$ is uniformly distributed on [0,1]. This result is a representation theorem of Choquet type, the uniform distributions being the extremal elements. The first two chapters introduce the necessary mathematical background via the setting of Khinchin spaces, basically the minimal requirements for a suitable Choquet representation via Radon measures. Chapter 3 discusses well-known concepts, convex, monotone, linear and Schur unimodality. Special attention is given to a new notion of beta unimodality, distributions with a Choquet representation via the beta distributions as extremals. In probabilistic notation, the distribution of $$UZ$$, where $$U$$ and $$Z$$ are independent and $$U$$ is beta distributed. The classical concept, treated in Chapter 4, is a special case. Chapter 5 is devoted to the discrete setting, basically in studying suitable Choquet representations. Chapter 6 deals with strong unimodality and Ibragimov-type results like unimodality preserving distributions. This and also the next chapter show also the importance of various orders, dispersive, convex, stochastic and slanted. Chapter 7 deals with the concept and consequences of slantedness. New insights to the mean-median-mode inequality for unimodal distributions are presented.
Reviewer: U.Rösler (Kiel)

##### MSC:
 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60E05 Probability distributions: general theory