Unimodality of Probability Measures

ISBN-13: 9789048147694 / Angielski / Miękka / 2010 / 256 str.

Emile M. J. Bertin; I. Cuculescu; Radu Theodorescu

Opis

Labor omnia vincit improbus. VIRGIL, Georgica I, 144-145. In the first part of his Theoria combinationis observationum erroribus min- imis obnoxiae, published in 1821, Carl Friedrich Gauss [Gau80, p.10] deduces a Chebyshev-type inequality for a probability density function, when it only has the property that its value always decreases, or at least does l not increase, if the absolute value of x increases . One may therefore conjecture that Gauss is one of the first scientists to use the property of 'single-humpedness' of a probability density function in a meaningful probabilistic context. More than seventy years later, zoologist W.F.R. Weldon was faced with 'double- humpedness'. Indeed, discussing peculiarities of a population of Naples crabs, possi- bly connected to natural selection, he writes to Karl Pearson (E.S. Pearson [Pea78, p.328]): Out of the mouths of babes and sucklings hath He perfected praise! In the last few evenings I have wrestled with a double humped curve, and have overthrown it. Enclosed is the diagram...If you scoff at this, I shall never forgive you. Not only did Pearson not scoff at this bimodal probability density function, he examined it and succeeded in decomposing it into two 'single-humped curves' in his first statistical memoir (Pearson [Pea94]).

Spis treści

Preface. 1. Prelude. 2. Khinchin Structures. 3. Concepts of Unimodality. 4. Khinchin's Classical Unimodality. 5. Discrete Unimodality. 6. Strong Unimodality. 7. Positivity of Functional Moments. Bibliography. Symbol Index. Name Index. Subject Index.