The Annals of Mathematical Statistics

On Minimizing and Maximizing a Certain Integral with Statistical Applications

Jagdish Sharan Rustagi

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

First available in Project Euclid: 27 April 2007

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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.

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