The Neumann-Forsythe method for simulation of an exponential family of distributions
The paper describes the Neumann-Forsythe method for the generation of exponentially distributed pseudo-random variables. Further, it contains a general statement and a quantitative analysis of this method for simulation of variables from the exponential family of distributions. Detailed algorithms for simulation from the exponential, normal and gamma distributions are obtained as special cases. The main result is the following Theorem concerning the correctness and analysis of the Neumann- Forsythe method and the exact algorithms illustrating the possibilities of the use of the described method. Theorem: Assume that the random value \(X_ 0\) has the c.d.f. \(F_ 0\), assume that \(X_ 1,X_ 2,\dots,\) which are independent of \(X_ 0\) and of each other, are uniformly distributed with the distribution \(F\). The following statements hold: a) if the integer value \(K \geq 1\) is determined by the condition \(X_ 0 \geq X_ 1 \geq \dots \geq X_{K-1} < X_ K\), then \[ P(X_ 0 \leq x\mid K\text{ odd}) = {1\over p}\int^ x_{-\infty}\text{exp}\{- F(y)\}dF_ 0(y) \] for any \(-\infty < x < \infty\), where \[ p = P(K\text{ odd}) = \int^ \infty_{-\infty}\text{exp}\{-F(y)\}dF_ 0(y). \] b) If \(N\) is the number of realizations of random values \(X_ 0,X_ 1,\dots,\) which are necessary to obtain one realization of the simulated random value \(X\), then \[ E(N) = {1\over p}(1+\int^ \infty_{- \infty}\text{exp}\{F(y)\}dF_ 0(y)), \] where \(1/p\) is the average number of series \(X_ 0 \geq X_ 1 \geq \dots\geq X_{k-1} < X_ K\) including the first odd \(K\).
exponential, normal and gamma distributions
scientific article; zbMATH DE number 482652