Fitting combinations of exponentials to probability distributions. (English) Zbl 1142.60321

In this paper two techniques for approximating distributions on the positive half-line by combinations of exponentials are described. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The class of approximating distributions considered in this paper consists of those with a distribution function \(F\) which can be written as \(1-F(t)=\sum_{j=1}^{n}a_{j}e^{-\lambda_{j}t}\), \(t\geq0, \lambda_{j}>0\), \(j=1,\dots,n\), \(1\leq n<\infty\). The author defines the Jacobi polynomials, gives some of their properties and shows how these polynomials may be used to obtain a convergent series of exponentials that represents a continuous distribution function exactly. Two versions of the method are described. The first one yields a convergent sequence of combinations of exponentials which are not precisely density functions, while the second version produces true density functions. It is shown that sequences of logbeta distributions converge to the Dirac mass, and that this can be used to find a different sequence of combinations of exponentials which converges to any distribution on a finite interval. The proposed methods are applied to approximation of some distributions: degenerate, uniform, Pareto, lognormal, Makeham and others. An error bound is given in the case of logbeta approximations.


60E99 Distribution theory
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