Use of Wilcoxon Test Theory in Estimating the Distribution of a Ratio by Monte Carlo Methods

Abstract

If $r = x/y$ is the ratio of two independent continuous positive random variables, its distribution can be estimated by generating random samples from the distribution of $x$ and $y$, and then proceeding in various ways. It is shown, using well-known results in the theory of Wilcoxon's test that the uniformly minimum variance unbiased estimate of $H(A) = P(r \leqq A)$ is obtained by computing Wilcoxon's statistic for the random variables $u_i = x_i, v_i = Ay_i(i = 1, \cdots, N)$. The variance of the estimate of $H(A)$ is readily estimated. The computations required by this approach are more arduous than those needed to estimate $H(A)$ from the quantities $r_i = x_i/y_i$, but may be worthwhile where the major part of the computations lies in generating the $x_i$ and $y_i$. Extension of the reasoning leads to choosing different numbers of $x$'s and $y$'s if they are of different complexity to generate. Further, if the distribution of one of the quantities $x$ or $y$ is known then an effectivity infinite sample from that population is already available and the distribution of $r$ can be estimated by sampling only the variable with unknown distribution, which may (or may not) result in economy of effort.