Journal of Mathematical Sciences

, Volume 139, Issue 3, pp 6497–6505 | Cite as

Student’s t-test for Gaussian scale mixtures

  • N. K. Bakirov
  • G. J. Székely


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.


Normal Random Variable Standard Normal Random Variable Exponential Power Scale Mixture Gaussian Scale Mixture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • N. K. Bakirov
    • 1
  • G. J. Székely
    • 2
  1. 1.Institute of MathematicsUfaRussia
  2. 2.Rényi Institute of MathematicsBudapestHungary

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