Student’s t-test for Gaussian scale mixtures
A Student-type test is constructed under a condition weaker than normal. We assume that 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 α, then the s-test provides the minimum critical values. (The most important critical values are tabulated at the end of the paper.) For α ≤.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 α. 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. Bibliography: 24 titles.
KeywordsNormal Random Variable Standard Normal Random Variable Exponential Power Scale Mixture Gaussian Scale Mixture
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- 4.H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1953).Google Scholar
- 12.H. Hotelling, “The behavior of some standard statistical tests under nonstandard conditions,” in: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley (1961), pp. 319–360.Google Scholar
- 14.J. Landerman, “The distribution of “Student’s” ratio for samples of two items drawn from non-normal universes,” Ann. Math. Statist., 10, 376–379 (1939).Google Scholar
- 23.G. J. Sz’ekely, Paradoxes in Probability Theory and Mathematical Statistics, Reidel, Dordrecht (1986).Google Scholar