Some convexity properties of the distribution of lower \(k\)-record values with extensions (Q2875242)

A distribution function (df) \(F\) is said to be unimodal if \(F\) is either convex, concave, or a value \(a\in\mathbb{R}\) exists such that \(F\) is convex on \((-\infty,a)\) and concave on \((a,\infty)\). A df \(F\) is said to be strongly unimodal if its convolution with any unimodal df is unimodal. It is well known that a non-degenerate df \(F\) is strongly unimodal if, and only if, it has a logconcave density \(f\).NEWLINENEWLINEIn this paper, it is shown that logconcavity of the reversed hazard rate \(r=f/F\) implies that of the density function. Using this result, unimodality properties of the df of lower \(k\)-records are derived.