## 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.

## Keywords

Normal Random Variable Standard Normal Random Variable Exponential Power Scale Mixture Gaussian Scale Mixture## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. S. Bartlett, “The effect of nonnormality on the
*t*-distribution,”*Proc. Camb. Phil. Soc.*,**31**, 223–231 (1935).zbMATHCrossRefGoogle Scholar - 2.S. Basu and A. DasGupta, “Robustness of standard confidence intervals for location parameters under departure from normality,”
*Ann. Statist.*,**23**, 1433–1442 (1995).zbMATHMathSciNetGoogle Scholar - 3.Y. Benjamini, “Is the
*t*-test really conservative when the parent distribution is long-tailed?, ”*J. Amer. Statist. Assoc.*,**78**, 645–654 (1983).zbMATHMathSciNetCrossRefGoogle Scholar - 4.H. Bateman and A. Erdelyi,
*Higher Transcendental Functions*, Vol. 1, McGraw-Hill, New York (1953).Google Scholar - 5.N. Cressie, “Relaxing assumption in the one-sided
*t*-test,”*Austral. J. Statist.*,**22**, 143–153 (1980).zbMATHMathSciNetCrossRefGoogle Scholar - 6.B. Efron, “Student’s
*t*-test under symmetry conditions,”*J. Amer. Statist. Assoc.*,**64**, 1278–1302 (1969).zbMATHMathSciNetCrossRefGoogle Scholar - 7.B. Efron and R. A. Olshen, “How broad is the class of normal scale mixtures?,”
*Ann. Statist.*,**6**, No. 5, 1159–1164 (1978).zbMATHMathSciNetGoogle Scholar - 8.C. Eisenhart, “On the transition of “Student’s”
*z*to “Student’s”*t*,”*Amer. Statist.*,**33**, No. 1, 6–10 (1979).MathSciNetCrossRefGoogle Scholar - 9.W. Feller,
*An Introduction to Probability Theory and Its Applications*, Vol. II, Wiley, New York-London-Sydney (1966).zbMATHGoogle Scholar - 10.R. A. Fisher, “Applications of “Student’s” distribution,”
*Metron*,**5**, 90–104 (1925).zbMATHGoogle Scholar - 11.T. Gneiting, “Normal scale mixtures and dual probability densities,”
*J. Stat. Comput. Simul.*,**59**, 375–384 (1997).zbMATHGoogle 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 - 13.D. Kelker, “Infinite divisibility and variance mixtures of the normal distribution,”
*Ann. Math. Statist.*,**42**, No. 2, 802–808 (1971).zbMATHMathSciNetGoogle 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 - 15.A. F. S. Lee and J. Gurland, “One-sample
*t*-test when sampling from a mixture of normal distributions,”*Ann. Statist.*,**5**, No. 4, 803–807 (1977).zbMATHMathSciNetGoogle Scholar - 16.A. V. Makshanov and O. V. Shalaevsky, “Some problems of asymptotic theory of distributions,”
*Zap. Nauchn. Semin. LOMI*,**74**, 118–138 (1977).zbMATHGoogle Scholar - 17.E. S. Pearson, “The distribution of frequency constants in small samples from nonnormal symmetrical and skew populations,”
*Biometrika*,**21**, 259–286 (1929).zbMATHCrossRefGoogle Scholar - 18.P. Prescott, “A simple alternative to Student’s
*t*,”*Appl. Statist.*,**24**, 210–217 (1975).MathSciNetCrossRefGoogle Scholar - 19.P. R. Rider, “On the distribution of the ratio of mean to standard deviation in small samples from nonnormal universes,”
*Biometrika*,**21**, 124–143 (1929).zbMATHCrossRefGoogle Scholar - 20.P. R. Rider, “On small samples from certain nonnormal universes,”
*Ann. Math. Statist.*,**2**, 48–65 (1931).zbMATHGoogle Scholar - 21.P. R. Rietz, “On the distribution of the “Student” ratio for small samples from certain nonnormal populations,”
*Ann. Math. Statist.*,**10**, 265–274 (1939).zbMATHMathSciNetGoogle Scholar - 22.
- 23.G. J. Sz’ekely,
*Paradoxes in Probability Theory and Mathematical Statistics*, Reidel, Dordrecht (1986).Google Scholar - 24.J. W. Tukey and D. H. McLaughin, “Less vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization,”
*Sankhyā, Ser. A*,**25**, 331–352 (1963).zbMATHGoogle Scholar