Is thetTest Really Conservative When the Parent Distribution is Long-Tailed?

Abstract

It is generally believed that the t test is conservative for a sample from a long-tailed symmetric distribution. Yet the probability inequalities expressing this property have not been proved. The inequalities are explored here using various criteria for long-tailedness and leaning on the geometrical interpretation of the t test. It is proved that the t test is conservative but only for large enough critical values. Examples of a liberal t test for lower values are given. The results are used to explain some curiosities in the asymptotic distribution of the t statistic and to study its behavior when the parent distribution is skewed.