Simultaneous assessment of normality and homoscedasticity in linear models

The validity of statistical inference relies heavily on the underlying assumptions. Normal distribution is very often assumed, especially in the context of analysis of variance (ANOVA). Another common
assumption that is made, once again in the context of ANOVA models, is
that of homoscedasticity. Formal testing of homoscedasticity is typically done under the normality assumption. A formal testing of normality is usually done assuming homoscedasticity. Thus, a simultaneous assessment of both of these assumptions is an important problem.

We utilize the concept of the Neyman smooth alternative, so that the
alternative hypothesis will reflect both non-normality and heteroscedasticity. Neyman’s smooth alternative provides a parametric family that is an alternative to normality; the normal distribution is a member of this family. The family is specified using a system of
orthogonal polynomials; Hermite polynomials and Legendre polynomials can both be used for this purpose. Under this family, we shall develop score tests that can simultaneously test normality and homoscedasticity, under
various statistical models, including some models that involve random effects. Selection of the order of the orthogonal polynomials can be data driven, and we derive such “data driven” tests. Our work involves univariate as well as some bivariate models. Extensive numerical results and illustrative examples will also be provided.