Computation of Confidence Limits for Linear Functions of the Normal Mean and Variance

Abstract

If a known transformation of a random variable X is normally distributed with mean {mu} and variance {sigma}{sup 2}, then the mean, variance, and any other distributional property of X can be expressed in terms of {mu} and {sigma}{sup 2}. For example, if X is lognormally distributed, i.e., X {approx} {Lambda}({mu}, {sigma}{sup 2}) or (equivalently) Y = log(X) {approx} N({mu}, {sigma}{sup 2}), then the expected value, variance, median, and mode of X are, respectively, E(X) = exp({mu} + {sigma}{sup 2}/2), var (X) = exp(2{mu} + {sigma}{sup 2})(exp({sigma}{sup 2}) - 1), med(X) = exp({mu}), and mode(X) = exp({mu} - {sigma}{sup 2}). Exact and optimal (uniformly most accurate unbiased) confidence limit procedures have been developed for linear functions of {mu} and {sigma}{sup 2} (Land, 1971, 1973) and, therefore, because confidence limits for a parameter are invariant under smooth, monotone transformations of that parameter, for the mean and mode of a lognormal distribution. In fact, the lognormal distribution is the only one whose mean can be expressed as a function of a non-trivial linear combination of {mu} and {sigma}{sup 2} (Land, 1971), but other functions, including those arising in connection with other normalizing transformations, can be approximated locally by linear functions for which exact limits can be constructed that define approximate limits for the original parametric functions of interest (Land, 1974, 1988). Tables have been published to facilitate the calculation of confidence limits for arbitrary linear functions of {mu} and {sigma}{sup 2} (Land, 1975), but their use is often tedious, requiring repeated interpolation and calculation. An unpublished Fortran program to compute confidence limits directly from sample estimates of p and u2 has been available from the second author, and has been used by a number of investigators to analyze lognormal data sets. The present paper introduces a more efficient computational algorithm, and documents the program for prospective users. An option has been added which makes it easy for the user to generate tables of confidence limits. Finally, the accuracy of the program has been thoroughly evaluated in terms of coverage probabilities for a wide range of parameter values.