Computing the Distribution of a Random Variable via Gaussian QuadratureRules

Computing the Distribution of a Random Variable via Gaussian Quadrature Rules By M. H. MEYERS (Manuscript received March 26, 1982) Using the technique of Gaussian quadrature rules, a new estimator is proposed for approximating the distribution of a random variable given only a finite number of its moments. The estimator is shown by numerous examples to be accurate on the tails of both continuous and discrete distributions. Efficient algorithms exist for computing the estimator from the first 2N moments of the random variable. A robust implementation of the estimator is presented, along with rules that provide additional protection against computer roundoff errors. I. INTRODUCTION In this paper we present a method for computing the Cumulative Distribution Function (CDF) of an arbitrary random variable. Using the theory of Gaussian Quadrature Rules (GQRS), we derive an estimator that converges asymptotically to the true CDF. In practice, convergence is obtained without excessive computation. A general estimator is developed here that is applicable to a wide class of problems. Section 2.1 begins with a review of GQR analysis as it has traditionally been used for numerical integration. Several authors have shown the existence of extremely efficient algorithms for computing the parameters of the GQR. An efficient and robust procedure for obtaining the GQR parameters is presented in the appendix. T w o CDF estimators based on GQR are derived in Sections 2.3 and 2.4. The first estimator is most suited to numerical integration schemes and estimation of discrete distributions, while the second is appropriate for continuous distributions such as Gaussian noise or crosstalk.