Some Numerical Aspects of the Use of Transforms in Statistics

Abstract

Abstract Numerical methods of differentiation and for the inversion of generating functions and Laplace transforms, are applied to some estimation problems. One problem considered involves the estimation of probabilities related to compound distributions. Let X 1, X 2, … be independent random variables with known distributions. Furthermore, let N be a random variable with an unknown distribution. The problem is to make inferences about the compound random variable Z = X 1 + X 2 + … + X N on the basis of k independent observations, N 1, N 2, …, N k , on N. Another problem considered involves the estimation of the busy period distribution in a Queueing system with known Poisson arrivals but unknown service distribution, on the basis of independent observations S 1, S 2, …, S k of the service time. The method uses a functional equation relating the Laplace transforms of the service time and busy period distributions. The same type of approach can be applied to a variety of estimation problems in Queueing and Renewal theory. Where the distribution of service time is known the method may be viewed as a numerical method for solving functional equations of the type frequently occurring in probability theory.