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

Abstract

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 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 s th and r th 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.