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June, 1957 On Minimizing and Maximizing a Certain Integral with Statistical Applications
Jagdish Sharan Rustagi
Ann. Math. Statist. 28(2): 309-328 (June, 1957). DOI: 10.1214/aoms/1177706961


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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Jagdish Sharan Rustagi. "On Minimizing and Maximizing a Certain Integral with Statistical Applications." Ann. Math. Statist. 28 (2) 309 - 328, June, 1957.


Published: June, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0088.35103
MathSciNet: MR88089
Digital Object Identifier: 10.1214/aoms/1177706961

Rights: Copyright © 1957 Institute of Mathematical Statistics

Vol.28 • No. 2 • June, 1957
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