Bounds for distribution functions of sums of squares and radial errors

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.