Efficiency of t-Test and Hotelling's T ²-Test After Box-Cox Transformation

Abstract

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 onesample 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.

Trend, periodicity and abrupt change in streamflow of the East River, the Pearl River basin

“…Thus, in this study, Lilliefors test (Lilliefors, ) was used to test the normality of monthly streamflow. If a time series failed to pass Lilliefors test, Box‐Cox transformation (Freeman and Modarres, ) would be applied to transfer the series to fit normal distribution. Also, as hydrological series are usually auto‐correlated, test statistics of scanning t and F tests were corrected by Table‐Look‐Up Test (Von and Zwiers, ; Jiang, ) to modify the significance criterion of lag‐1 autocorrelation coefficients.…”

The day‐of‐the‐week effect and conditional volatility: Sensitivity of error distributional assumptions

“…66Violations of non‐normality such as those arising from skewness and/or excess kurtosis in the population introduce biases in traditional hypothesis testing. For example, Freeman and Modarres (2006) note that with a skewed distribution the actual p‐values can be larger than the values calculated from the t ‐tables. Early investigations of the effects of non‐normality by Neyman and Pearson (1928) show that skewness has a greater effect on the distribution of t‐statistics than kurtosis but heavy tails still produce bias.…”