Two noncentral chi-square generalizations

Generalizations of two known noncentral chi-square results are presented. The first generalization concerns extending the expression for the probability that one (first-order) Ricean random variable exceeds another to the expression that one nth-order Ricean random variable exceeds another nth-order Ricean random variable. This latter result is shown to be equivalent to the probability of one noncentral chi-square random variable with 2n degrees of freedom exceeding another. The form of the resulting expression is such that it can easily be evaluated by a recurrence relation. The second generalization deals with the fact that the noncentral chi-square distribution function, with d degrees of freedom, differs from the complementary probability of detection, 1-P_N(X, Y), only in that the latter is restricted to even degrees of freedom. We can expand the techniques and expressions developed for the robust, efficient, and accurate calculation of the P _N(X, Y), or equivalently, the generalized Marcum (1960) Q-function, to the calculation of the noncentral chi-square distribution function and its complement