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 <disp-formula><tex-math><![CDATA[$$\begin{eqnarray} F^{(1)}(t) &=& F(t),\\ F^{(n+1)}(t) &=& \int^t_0 F^{(n)}(t-x)F'(x)\,dx \quad n\geq 1,\end{eqnarray}$$]]></tex-math></disp-formula> 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 \int ₀^tP^(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.