[Paper Review] Mills' ratio: Reciprocal concavity and functional inequalities

Árpád Baricz|arXiv (Cornell University)|Oct 15, 2010

Bayesian Methods and Mixture Models11 references3 citations

TL;DR

This paper establishes sufficient conditions for the Mills ratio of continuous univariate distributions to be reciprocally convex or concave, using properties like complete monotonicity and the monotone form of l’Hôpital’s rule. It derives functional inequalities for the Mills ratios of gamma, log-normal, and Student’s t distributions, with applications in monopoly theory.

ABSTRACT

AbstractThis note contains sufficient conditions for the probability density function of an arbitrary continuousunivariate distribution such that the corresponding Mills ratio to be reciprocally convex (concave). Toillustrate the applications of the main results, the Mills ratio of some common continuous univariate dis-tributions, like gamma, log-normal and Student’s t distributions, are discussed in details. The applicationto monopoly theory is also summarized. Keywords: Mills ratio, Reciprocally convex (concave) functions, Monotone form of l’Hospital’s rule,Statistical distributions, Completely monotonic functions, Stieltjes transform1. IntroductionBy definition (see [Me]) a function f : [a,b] ⊆ (0,∞) → Ris said to be (strictly) reciprocally convexif x → f(x) is (strictly) concave and x → f(1/x) is (strictly) convex on [a,b]. Merkle [Me] showed thatf is reciprocally convex if and only if for all x,y ∈ [a,b] we havef2xyx +y≤f(x) + f(y)2≤ fx+y2≤xf(x) +yf(y)x+y. (1)We note here that in fact the third inequality follows from the fact that the function x → f(1/x) is convexon [a,b] if and only if x → xf(x) is convex on [a,b]. In what follows, a function g : [a,b] ⊆ (0,∞) → Rissaid to be (strictly) reciprocally concave if and only if −g is (strictly) reciprocally convex, i.e. if x → g(x)is (strictly) convex and x → g(1/x) is (strictly) concave on [a,b]. Observe that if f is differentiable, thenx → f(1/x) is (strictly) convex (concave) on [a,b] if and only if x → x

Motivation & Objective

• To identify sufficient conditions under which the Mills ratio of a continuous univariate distribution is reciprocally convex or concave.
• To analyze the Mills ratios of common distributions—gamma, log-normal, and Student’s t—using functional inequalities.
• To apply the theoretical results to economic models, particularly monopoly theory.
• To extend the use of the monotone form of l’Hôpital’s rule and completely monotonic functions in studying Mills ratios.
• To establish connections between reciprocal convexity and the Stieltjes transform in distributional analysis.

Proposed method

• Utilizes the definition of reciprocally convex functions: f is reciprocally convex if f(1/x) is convex and f(x) is concave on [a,b].
• Applies the monotone form of l’Hôpital’s rule to analyze the monotonicity of ratios of functions, especially in the context of Mills ratios.
• Employs the concept of completely monotonic functions to derive conditions under which the Mills ratio exhibits reciprocal convexity.
• Uses the inequality chain f(2xy/(x+y)) ≤ (f(x)+f(y))/2 ≤ f((x+y)/2) ≤ (xf(x)+yf(y))/(x+y) as a characterization of reciprocal convexity.
• Analyzes the transformation x → xf(x) to relate convexity of f(1/x) to convexity of xf(x), enabling functional analysis.
• Applies the Stieltjes transform framework to represent and study the Mills ratio in terms of integral representations.

Experimental results

Research questions

• RQ1 Under what conditions is the Mills ratio of a continuous univariate distribution reciprocally convex or concave?
• RQ2 How do the functional inequalities involving the harmonic and arithmetic means characterize reciprocal convexity of the Mills ratio?
• RQ3 What are the implications of reciprocal convexity for the gamma, log-normal, and Student’s t distributions?
• RQ4 How can the monotone form of l’Hôpital’s rule be used to analyze the monotonicity of the Mills ratio?
• RQ5 In what way does reciprocal convexity of the Mills ratio impact models in monopoly theory?

Key findings

• The Mills ratio of the gamma distribution is reciprocally convex under specific parameter constraints, as shown via complete monotonicity.
• For the log-normal distribution, the Mills ratio is reciprocally convex when the underlying normal distribution has a sufficiently large variance.
• The Mills ratio of the Student’s t distribution is reciprocally convex for degrees of freedom greater than a certain threshold, derived using asymptotic and monotonicity analysis.
• The inequality chain f(2xy/(x+y)) ≤ (f(x)+f(y))/2 ≤ f((x+y)/2) ≤ (xf(x)+yf(y))/(x+y) holds for reciprocally convex Mills ratios, providing a functional characterization.
• The transformation x → xf(x) preserves convexity when f(1/x) is convex, enabling the use of convexity criteria in reciprocal analysis.
• The application to monopoly theory reveals that reciprocal convexity of the Mills ratio influences the structure of optimal pricing strategies under asymmetric information.