International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1991 / Article

Open Access

Volume 14 |Article ID 292756 | https://doi.org/10.1155/S0161171291000765

Roger B. Nelsen, Berthold Schweizer, "Bounds for distribution functions of sums of squares and radial errors", International Journal of Mathematics and Mathematical Sciences, vol. 14, Article ID 292756, 9 pages, 1991. https://doi.org/10.1155/S0161171291000765

Bounds for distribution functions of sums of squares and radial errors

Received19 Oct 1990
Revised19 Feb 1991

Abstract

Bounds are found for the distribution function of the sum of squares X2+Y2 where X and Y are arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible when X and Y are symmetric about 0 and yield expressions which can be evaluated explicitly when X and Y have a common distribution function which is concave on (0,). Similar results are obtained for the radial error (X2+Y2)½. The important case where X and Y are normally distributed is discussed, and here best-possible bounds on the circular probable error are also obtained.

Copyright © 1991 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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