ELSEVIER Statistics Probability Letters

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ELSEVIER Statistics & Probability Letters 28 (1996) 367-373 Tailweight with respect to the mode for unimodal distributions J, Averous, A.-L. Fougbres *, M. Meste Laboratoire de Statistique t Probabilit~s, Universit~ Paul Sabatier, 118 Rte de Narbonne, 31062 Toulouse cedex, France Received November 1993; revised December 1994 Abstract Location, spread, skewness and tailweight are studied for unimodal distributions by means of mode-based concepts. The Lrvy concentration function and notions related to it are playing an important part. AMS Subject Classification: Primary 62E10; Secondary 60E99 Keywords: Mode; Concentration function; Location; Spread; Skewness, Tailweight I. Introduction Unimodal distributions form a remarkable subset of probability distributions, which presents nice properties. Several characterizations of unimodal distributions can be found in the literature: concentration functions (Benin et al., 1981), characteristic functions (e.g. Dharmadhikari and Joag-Dev, 1988, p. 7). In Bertin et al.'s approach to unimodality, the notions of concentration function QF and pointer AF of a distribution function F are essential. Location, scale, skewness and tailweight are important concepts for the description of a probability distribu- tion. The study of tails for skewed distributions presents a particular aspect for unimodal distributions: when we consider the graph of the probability density function, the mode seems to be an appealing centre.