Student's $t$-test for Gaussian scale mixtures

A Student type test is constructed under weaker than normal condition. We assume the errors are scale mixtures of normal random variables and compute the critical values of the suggested $s$-test. Our $s$-test is optimal in the sense that if the level is at most $\alpha$, the $s$-test provides the minimal critical values. (The most important critical values are tablulated at the end of the paper.) For $\alpha\le.05$ the two-sided $s$-test is identical with Student's classical $t$-test. In general, the $s$-test is a $t$-type test but its degree of freedom should be reduced depending on $\alpha$. The $s$-test is applicable for many heavy tailed errors including symmetric stable, Laplace, logistic, or exponential power. Our results explain when and why the $P$-value corresponding to the $t$-statistic is robust if the underlying distribution is a scale mixture of normal distributions.