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# On Minimizing and Maximizing a Certain Integral with Statistical Applications

Jagdish Sharan Rustagi
The Annals of Mathematical Statistics
Vol. 28, No. 2 (Jun., 1957), pp. 309-328
Stable URL: http://www.jstor.org/stable/2237155
Page Count: 20
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## 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.

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