On the concavity of multivariate probability distribution functions

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Abstract

We prove that the multivariate standard normal probability distribution function is concave for large argument values. The method of proof allows for the derivation of similar statements for other types of multivariate probability distribution functions too. The result has important application, e.g., in probabilistic constrained stochastic programming problems.

Introduction

Let Φ(z;R) be the n-variate standard normal probability distribution function with correlation matrix R. We assume that R is nonsingular, i.e., the distribution is nondegenerate.

It is well known (see, e.g., [3]) that Φ(z;R) is logarithmically concave (logconcave) in the entire space Rn.

Logconcave probability distributions have many important applications. In probabilistic constrained stochastic programming we use them to prove the convexity of a large class of problems. If we have, e.g., a constraint of the formP(Tx⩾X)⩾p,where X is a normally distributed random vector, then the logconcavity property of its distribution function implies that the set of x vectors satisfying (1.1) is a convex set.

In some applications the distribution of the random vector X is a mixture of normal distributions (e.g., in multiple decrement insurance problems) which is not logconcave, in general. In fact, logconcavity does not carry over from terms to their sum. However, concavity does carry over, hence it is important to look for this property in connection with normal as well as other multivariate probability distribution functions.

The main purpose of this paper is to prove that if the components of z are large, then Φ(z;R) is not only logconcave but also concave. The method of proof can be applied to other probability distribution functions too, as we show it in Section 3, in case of the Dirichlet distribution.

Section snippets

The Main Theorem

In this section we prove the following

Theorem 2.1

The function Φ(z1,…,zn;R) is concave in the set {z|zin−1,i=1,…,n}.

Proof

If Z has distribution function Φ(z;R), then it can be represented as Z=AV, where A is a matrix that satisfies R=AAT and V is a random vector that has independent, standard normally distributed components.

We have the equalityΦ(z;R)=P(AV⩽z)=∫⋯Av⩽zφ(v1)⋯φ(vn)dv1dvn,where φ is the probability density function of the standard normal distribution.

Let us introduce polar coordinates in the

Miscellaneous remarks

The lower bound (in Theorem 2.1) n−1 for the components of z to ensure concavity of Φ(z;R) is not always the best one. If, e.g., ρ=0, then the bivariate standard normal probability distribution function is concave for, zi⩾0.51, i=1,2. In fact, the Jacobian of −Φ(z1)Φ(z2) is positive definite if ziΦ(zi)/φ(zi)>1 and it holds for zi⩾0.51, i=1,2. It is an open problem to determine that α=α(R) for which Φ(z;R) is concave in the set {z|zi⩾α,i=1,…,n} and then find the worst bound, i.e., the maximum of

Acknowledgements

I express my thanks to the referee for his/her helpful remarks, especially for calling my attention that the proof of Theorem 2.1 can be shortened.

References (3)

  • T.W. Anderson

    An Introduction to Multivariate Statistical Analysis

    (1984)
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