Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

Zakhar Kabluchko; Axel Munk

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The proportional likelihood ratio order and applications.

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In this paper, we introduce a new stochastic order between continuous non-negative random variables called the PLR (proportional likelihood ratio) order, which is closely related to the usual likelihood ratio order. The PLR order can be used to characterize random variables whose logarithms have log-concave (log-convex) densities. Many income random variables satisfy this property and they are said to have the IPLR (increasing proportional likelihood ratio) property (DPLR property)....