A Numerical Algorithm for Recursively-Defined Convolution Integrals Involving Distribution Functions

Reliability studies give rise to families of distribution functions F (n) defined recursively by the repeated convolution of a distribution function F with itself according to the scheme [Formula: see text] where F′ is the derivative of F, and is usually given by a p.d.f. f. In particular, many systems characteristics are defined in terms of integrals of the form ∫ 0 t P (s) (t − x)Q (r) (x) dx where P (s) and Q (r) are the sth and rth members of families generated from distribution functions P and Q, not necessarily distinct. It is seldom possible or convenient to express the F (n) in analytical form. An algorithm based on cubic spline interpolation is given here for recursively generating continuous numerical approximations to the F (n) in a form which allows them to be convoluted together to provide useful approximation to the second of the above integrals.