## Abstract

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.

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Published in *Zapiski Nauchnykh Seminarov POMI*, Vol. 328, 2005, pp. 5–19.

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Bakirov, N.K., Székely, G.J. Student’s *t*-test for Gaussian scale mixtures.
*J Math Sci* **139**, 6497–6505 (2006). https://doi.org/10.1007/s10958-006-0366-5

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DOI: https://doi.org/10.1007/s10958-006-0366-5