Joint Probability Density Function of Selected Order Statistics and the Sum of the Remaining Random Variables

A set of N independent, identically distributed random variables Xsub n, with common probability density function px, are ordered into a new set of dependent random variables Xsub n, each with a different probability density function. From this latter set, the n1-th largest random variable through the nsub M-1-th largest random variable are selected. Then, the sum of the remaining N1-M random variables is computed, giving a total of M dependent random variables. The joint probability density function of these M random variables is derived in a form involving a single Bromwich contour integral in the moment-generating function domain. The integral is most easily numerically evaluated by locating approximately the real saddlepoint of the integrand and passing the contour through this point. Very high accuracy in the probability density function evaluation is available by using numerical integration instead of a saddlepoint approximation.