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Bounds for distribution functions of sums of squares and radial errors

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  • Roger B. Nelsen
  • Berthold Schweizer

Abstract

Bounds are found for the distribution function of the sum of squares X 2 + Y 2 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 ( X 2 + Y 2 ) 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.

Suggested Citation

  • Roger B. Nelsen & Berthold Schweizer, 1991. "Bounds for distribution functions of sums of squares and radial errors," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 14, pages 1-9, January.
    • Handle: RePEc:hin:jijmms:292756
    DOI: 10.1155/S0161171291000765
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