## The Annals of Mathematical Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

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**

links.jstor.org

#### Citation

Rustagi, Jagdish Sharan. On Minimizing and Maximizing a Certain Integral with Statistical Applications. Ann. Math. Statist. 28 (1957), no. 2, 309--328. doi:10.1214/aoms/1177706961. https://projecteuclid.org/euclid.aoms/1177706961