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
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.
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