Unimodality of Probability Measures

1. Auflage. Softcover version of original hardcover Auflage 1997
ISBN: 978-90-481-4769-4
Verlag: Springer

Buch, Englisch, 256 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 417 g

Reihe: Mathematics and Its Applications

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]).

Zielgruppe

Research

Autoren/Hrsg.

Bertin, Emile M.J.

Cuculescu, I.

Theodorescu, Radu

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Stochastik Elementare Stochastik
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Datenanalyse, Datenverarbeitung
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung

Inhaltsverzeichnis

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.- Symbol index.- Name index.