The Mean of the Inverse of a Punctured Normal Distribution and Its Application

Summary The fundamental properties of a punctured normal distribution are studied. The results are applied tothree issues concerning X=Y where X and Y are independent normal random variables with means m X and m Y respectively. First, estimation of m X =m Y as a surrogate for EðX=YÞ is justified, then the reasonfor preference of a weighted average, over an arithmetic average, as an estimator of m X =m Y is given.Finally, an approximate confidence interval for m X =m Y is provided. A grain yield data set is used toillustrate the results. Key words: Arithmetic average; Coefficient of variation; Inverse mean; Punctured normal;Ratio; Truncated; Weighted average. 1 Introduction In agricultural research, sample surveys and other areas, one is often concerned with the ratioR ¼ X=Y of two independent normal random variables, X Nðm X ;s 2X Þ and Y Nðm Y ;s 2Y Þ wherem X ;m Y > 0. Properties of the distribution of this ratio, in particular its moments, are of interest toresearchers. The mean EðX=YÞ of the ratio of two independent normal variables, however, does notexist (see for example, Springer, 1979, p. 139). The root of the problem is the non-existence ofEð1=YÞ; this occurs because Y can in theory assume values arbitrarily close to zero.In order to resolve a number of issues surrounding this problem, in this paper we study Y forjYj > e, with e a small positive number, a “punctured normal” distribution; a small neighbourhood ofzero is removed from consideration. We examine the fundamental properties of this distribution, inparticular we show that the inverse mean of a punctured normal does exist and an explicit expressionis obtained. Approximations for Eð1=YÞ have been developed for a left-truncated normal random vari-able Y (Nahmias and Wang, 1978; Hall, 1979), but the expression for the inverse mean of a puncturednormal given in this paper is exact.We then apply our results to three issues surrounding R ¼ X=Y. First, we justify estimation ofm