AN EXACT TRANSFORMATION LINKING LOGNORMAL AND CHI-SQUARE DISTRIBUTIONS

Abstract

The lognormal and chi-square distributions come from complementary mechanisms: multiplicative accumulation, and sums of squared Gaussian deviates. One elementary link between them is well known — the square of a standard normal variable is chi-square with one degree of freedom. This link has rarely been formalized into a general, ready-to-use framework. No single source generalizes it to noncentral and heterogeneous-parameter settings, verifies it by simulation, and packages it into usable inferential tools. That is the gap this paper addresses. We prove three theorems. Theorem 1 shows that logarithmic standardization of k independent lognormal variates yields an exact chi-square pivot with k degrees of freedom. Theorem 2 shows that misspecifying the assumed location parameter shifts this pivot to a noncentral chi-square law, with noncentrality proportional to the squared standardized bias. Theorem 3 shows that heterogeneous lognormal families with unequal dispersion parameters map onto a generalized, weighted chi-square variable that reduces to the ordinary case when dispersions coincide. We also give a reverse construction that generates lognormal variates from chi-square draws. Extensive Monte Carlo simulation (200,000 replicates per configuration) confirms all three results to high precision. Kolmogorov–Smirnov tests fail to reject the theoretical laws in every configuration examined. We demonstrate the framework with two applications: a diagnostic test for a hypothesized lognormal location, and an exact 95% confidence interval for the lognormal dispersion parameter σ², both built directly from the chi-square identity. The framework offers exact, finite-sample inference tools for fields where the lognormal model is standard, including reliability engineering, environmental monitoring, and financial volatility analysis.