Computing Tail Probabilities by Numerical Fourier Inversion: the Absolutely Continuous Case

The numerical computation of P{X > x} can be accomplished in a variety of ways. An appealing class of methods may be derived from the contour integral connecting P{X > x} and its Fourier representation. Statisticians have largely focused on deriving saddlepoint approximations for this contour integral. The accuracy of such approximations is generally understood in vague terms only and, perhaps more importantly, is rarely under user control. Numerical integration of the contour integral has received considerably less attention, particularly in the statistics literature. The focus of this paper is on the use of the trapezoidal rule applied to said contour integral along an appropriate path. An exponential bound on the approximation (i.e., discretization) error of the trapezoidal rule as a function of the quadrature node spacing is obtained using results of Stenger (1993). This bound is used in developing a reliable non-iterative method of selecting the trapezoidal rule spacing that guarantees control of the approximation error. The epsilon algorithm is used to accelerate the calculation of the tail of the infinite series that results upon applying the trapezoidal rule to the inversion integral. The resulting “automatic” methodology is shown to produce extremely accurate results in a diverse set of problems.