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Bounds for distribution functions of sums of squares and radial errors

Roger B. Nelsen, Berthold Schweizer
1991 International Journal of Mathematics and Mathematical Sciences  
Bounds are found for the distribution function of the sum of squaresX2+Y2whereXandYare arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible whenXandYare symmetric about0and yield expressions which can be evaluated explicitly whenXandYhave a common distribution function which is concave on(0,∞). Similar results are obtained for the radial error(X2+Y2)½. The important case whereXandYare
more » ... ereXandYare normally distributed is discussed, and here best-possible bounds on the circular probable error are also obtained.
doi:10.1155/s0161171291000765 fatcat:p63b43qehbeprgjpnvfvgra2ei