International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 561-569
Bounds for distribution functions of sums of squares and radial errors
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.
Bounds are found for the distribution function of the sum of squares where and
are arbitrary continuous random variables. The techniques employed, which utilize copulas and their
properties, show that the bounds are pointwise best-possible when and are symmetric about and
yield expressions which can be evaluated explicitly when and have a common distribution function
which is concave on . Similar results are obtained for the radial error . The important
case where and are normally distributed is discussed, and here best-possible bounds on the circular
probable error are also obtained.