On Minimizing and Maximizing a Certain Integral with Statistical Applications

Abstract

We consider here the problem of minimizing and maximizing $\int^x_{-x\varphi}(x, F(x)) dx$ under the assumptions that $F(x)$ is a cumulative distribution function (cdf) on $\lbrack -X, X\rbrack$ with the first two moments given and that $\varphi$ is a certain known function having certain properties. The existence of the solution has been proved and a characterization of the maximizing and minimizing cdf's given. The minimizing cdf is unique when $\varphi(x, y)$ is strictly convex in $y$ and is completely characterized for some special forms of $\varphi$. The maximizing cdf is a discrete distribution and in the above case turns out to be a three-point distribution. Several statistical applications are discussed.