## The Annals of Mathematical Statistics

### On Minimizing and Maximizing a Certain Integral with Statistical Applications

Jagdish Sharan Rustagi

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 28, Number 2 (1957), 309-328.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177706961

Digital Object Identifier
doi:10.1214/aoms/1177706961

Mathematical Reviews number (MathSciNet)
MR88089

Zentralblatt MATH identifier
0088.35103

JSTOR