Saddlepoint approximations: a review and some new applications

The saddlepoint method of approximation is attributed to Daniels (1954), and can be described in basic terms as yielding an accurate and usually fast and very numerically reliable approximation to the mass or density function (hereafter pdf), and the cumulative distribution function (cdf), of a random variable, say X, based on knowledge of its moment generating function (mgf). Denote the latter by M X (s) , where s is the real argument of the function, such that s is contained in the convergence strip of M X (s) , to be defined below. Several surveys and monographs are available; the best starting point is the currently definitive exposition in Butler (2007), along with the first textbook dedicated to the subject, Jensen (1995). Our goal is to outline the basics of the methodology in the easiest way possible, and then to illustrate a small subset of its many applications.