Multiplicative Strong Unimodality

Abstract

Multiplicative strong unimodality is defined as the preservation of unimodality in products of independent random variables. An Ibragimov type theorem is proved. As an application, preserving unimodality for scale mixtures of gamma distributions is examined. It is also shown that multiplicative strong unimodal probability measures on IR appear as images, by the exponential map, of classical strong unimodal ones. The connection to the star order is also established.

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

“…[13] yields logconcavity of g r . Further, we can apply Lemma 4.5 in Cramer [13] in combination with the notion of multiplicative strong unimodality as introduced by Cuculescu & Theodorescu [16]. This proves that the cdf of a uniform DGOS is always unimodal (cf.…”

Logconcavity and unimodality of progressively censored order statistics

“…Therefore, we establish unimodality of gOSs by the notion of multiplicative strong unimodality (cf. Cuculescu and Theodorescu, 1998). In particular, this property implies unimodality of the underlying cdf.…”

On the unimodality of power transformations of positive stable densities

“…In general, techniques for investigating multimodality are different from those devoted to unimodality. Anyway it does not seem that the methods of this paper, which in the non‐monotonic situation rely on the notion of strong multiplicative unimodality 3, can be of any help for proving bimodality. It would also be interesting to study the unimodality of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$X_\alpha ^r,$\end{document} where X α is a general α‐stable variable conditioned to stay positive.…”

“…On the other hand, one sees explicitly that the function x ↦λ α, r ( e x ) is log‐concave for all α ∈ (0, 1) and r ≠ 0, where λ α, r is the density of L r (α − 1)/α . By Theorem 3.7. in 3, this means that L r (α − 1)/α is multiplicatively strong unimodal, in other words, that its independent product with any unimodal random variable remains unimodal. Recalling we can deduce that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$Z_\alpha ^r\in \mathcal U$\end{document} as soon as r ≥ −α.…”