International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 561-569
doi:10.1155/S0161171291000765

Bounds for distribution functions of sums of squares and radial errors

Roger B. Nelsen1 and Berthold Schweizer2

1Department of Mathematics, Lewis and Clark College, Portland 97219, Oregon, USA
2Department of Mathematics and Statistics, University of Massachusetts, Amherst 01003, Massachusetts, USA

Received 19 October 1990; Revised 19 February 1991

Copyright © 1991 Roger B. Nelsen and Berthold Schweizer. 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.

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.