Saddlepoint Approximations for the Combined Probability and Joint Probability Density Function of Selected Order Statistics and the Sum of the Remainder

A set of N independent random variables ?Xn! with arbitrary probability density functions (pn(X)! are ordered into a new set of dependent random variables, each with a different probability density function. From this new set, the M - I largest random variables are selected. Then, the sum of the remaining N + I - M random variables is computed, giving a total of M dependent random variables. Several statistics have been computed for these M random variables, including their joint probability density function and a quantity called the "combined probability and joint probability density function" of particular selections. Most of the results can be expressed in terms of a single Bromwich contour integral in the moment-generating domain. A saddlepoint approximation and first-order correction term are derived for this contour integral and then applied to several typical examples. Actual numerical calculation of these saddlepoint approximations requires evaluation of several functions that have high-order removable singularities, thereby requiring power series expansions near these removable singularities. Expansions with 13 to 14 decimal-digit accuracy have been derived and programmed for this purpose.