A spline-based algorithm for approximation of the n-fold convolution of the probability density functions

Abstract

In this dissertation an algorithm is developed to approximate the n-fold convolution integral of the probability density functions which are analytic over some interval I = {a,b}. The algorithm uses spline approximation method to interpolate the underlying density functions. The interpolation is performed over the interval I by using a uniform partition. Once the spline approximation of the density functions is found, the convolution of the resulting spline functions is found analytically. Two different schemes of approximation are considered and their performance is compared against one another. One scheme uses simple knots while the other scheme uses multiple knots. In scheme 2, five new spline functions of eighth-order are created. The algorithm developed in this work is applied to three different problems in inventory control, renewal processes, and reliability engineering. FORTRAN routines are written for each problem.