Welch’s ANOVA: Heteroskedastic skew-t error terms

Abstract

In analysis of variance (ANOVA) models, it is generally assumed that the distribution of the error terms is normal with mean zero and constant variance ${\sigma }^2.$ Traditionally, a least square (LS) method is used for estimating the unknown parameters and testing the null hypothesis. It is known that, when the normality assumption is not satisfied, LS estimators of the parameters and the test statistics based on them lose their efficiency, see Tukey. On the other hand, a non constant variance problem which is called heteroskedasticity is another serious issue for LS estimators. The LS estimators still remain unbiased but the estimated standard error (SE) is wrong. Because of this, confidence intervals and hypotheses tests cannot be relied on. Welch’s ANOVA is the most popular method to solve this problem. In this paper, we assume that the distribution of the error terms is skew-t with non constant variance in one-way ANOVA. We also propose a new test statistics based on the maximum likelihood (ML) estimators of skew-t distribution. A Monte Carlo simulation study is performed to compare traditional LS and Welch’s method with the proposed method in terms of Type I errors and the powers. At the end of this study, an example is given for the illustration of the methods.

Keywords:

• Heteroskedasticity
• Welch’s ANOVA
• skew-t distribution
• maximum likelihood