Revisiting the critical values of the Lilliefors test: towards the correct agrometeorological use of the Kolmogorov-Smirnov framework

Abstract

Several studies have applied the Kolmogorov-Smirnov test (KS) to verify if a particular parametric distribution can be used to assess the probability of occurrence of a given agrometeorological variable. However, when this test is applied to the same data sample from which the distribution parameters have been estimated, it leads to a high probability of failure to reject a false null hypothesis. Although the Lilliefors test had been proposed to remedy this drawback, several studies still use the KS test even when the requirement of independence between the data and the estimated parameters is not met. Aiming at stimulating the use of the Lilliefors test, we revisited the critical values of the Lilliefors test for both gamma (gam) and normal distributions, provided easy-to-use procedures capable of calculating the Lilliefors test and evaluated the performance of these two tests in correctly accepting a hypothesized distribution. The Lilliefors test was calculated by using critical values previously presented in the scientific literature (KSLcrit) and those obtained from the procedures proposed in this study (NKSLcrit). Through Monte Carlo simulations we demonstrated that the frequency of occurrence of Type I (II) errors associated with the KSLcrit may be unacceptably low (high). By using the NKSLcrit we were able to meet the significance level in all Monte Carlo experiments. The NKSLcrit also led to the lowest rate of Type II errors. Finally, we also provided polynomial equations that eliminate the need to perform statistical simulations to calculate the Lilliefors test for both gam and normal distributions.