A note on the misuses of the variance test in meteorological studies

Abstract

Stochastic modeling of rainfall data is an important area in meteorology. The gamma distribution is a widely used probability model for non-zero rainfall. Typically the choice of the distribution for such meteorological studies is based on two goodness-of-fit tests—the Pearson’s Chi-square test and the Kolmogorov–Smirnov test. Inspired by the index of dispersion introduced by Fisher (Statistical methods for research workers. Hafner Publishing Company Inc., New York, 1925), Mooley (Mon Weather Rev 101:160–176, 1973) proposed the variance test as a goodness-of-fit measure in this context and a number of researchers have implemented it since then. We show that the asymptotic distribution of the test statistic for the variance test is generally not comparable to any central Chi-square distribution and hence the test is erroneous. We also describe a method for checking the validity of the asymptotic distribution for a class of distributions. We implement the erroneous test on some simulated, as well as real datasets and demonstrate how it leads to some wrong conclusions.