A generalized normal scores test that increases the power of a test of significance for a coefficient in a linear model

A generalization of the normal scores two-sample test is proposed. The empirical significance level of this test closely approximates the nominal level of significance and it is often more powerful than the traditional t test for a test of any single coefficient in a linear model. Based on an extensive simulation study we found that, for most applications, the generalized normal scores test maintains its level of significance and it has, for data sets having 50 or more observations, greater power than the t test if the distribution of the errors is long-tailed or skewed. But, we found that the empirical significance level slightly exceeds the nominal level for regression models having covariates that are skewed and highly correlated, and for those models the generalized normal scores test is not recommended. If a data set has fewer than 50 observations the generalized normal scores test is not recommended because for most error distributions it does not offer an increase in power over the t test. However, if the number of observations equals or exceeds 50 and the covariates are not highly correlated this generalized normal scores test can be used to increase the power of tests of significance.