Efficiency of t-test and Hotelling's T2-test after Box-Cox transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t -statistic than does kurtosis. When the distribution is skewed, the actual p -values can be larger than the values calculated from the t -tables. Transformation of data to normality has shown good results in the case of univariate t -test. In order to reduce the effect of skewness of the distribution on normal-based t -test, one can transform the data and perform the t -test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t -test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t -test and Hotelling's T-2-test of multivariate location with respect to the same statistic based on untransformed data is at least one.