Mills' ratio: Reciprocal convexity and functional inequalities

TL;DR

This paper establishes conditions under which the Mills ratio of a continuous distribution on (0,∞) is reciprocally convex or concave, with detailed application to the gamma distribution.

Contribution

It provides new sufficient conditions for reciprocal convexity or concavity of Mills ratios, extending understanding of their functional inequalities.

Findings

Derived conditions for reciprocal convexity of Mills ratio.

Applied results to gamma distribution's Mills ratio.

Enhanced theoretical understanding of Mills ratio properties.

Abstract

This note contains sufficient conditions for the probability density function of an arbitrary continuous univariate distribution, supported on $(0,\infty),$ such that the corresponding Mills ratio to be reciprocally convex (concave). To illustrate the applications of the main results, the reciprocal convexity (concavity) of Mills ratio of the gamma distribution is discussed in details.