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Student's $t$-test for Gaussian scale mixtures
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 5-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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N. K. Bakirov; G. J. Szekely. Student's $t$-test for Gaussian scale mixtures. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 5-19. http://geodesic.mathdoc.fr/item/ZNSL_2005_328_a0/

[1] M. S. Bartlett, “The effect of nonnormality on the $t$-distribution”, Proc. Camb. Phil. Soc., 31 (1935), 223–231 | DOI

[2] S. Basu, DasGupta, “Robustness of standard confidence intervals for location parameters under departure from normality”, Ann. Statist., 23 (1995), 1433–1442 | DOI | MR | Zbl

[3] Y. Benjamini, “Is the $t$-test really conservative when the parent distribution is long-tailed?”, J. Amer. Statist. Assoc., 78 (1983), 645–654 | DOI | MR | Zbl

[4] H. Bateman, A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953 | MR

[5] N. Cressie, “Relaxing assumption in the one-sided $t$-test”, Australian J. Statist., 22 (1980), 143–153 | DOI | MR | Zbl

[6] B. Efron, “Student's $t$-test under symmetry conditions”, J. Amer. Statist. Assoc., 64 (1969), 1278–1302 | DOI | MR | Zbl

[7] B. Efron, R. A. Olshen, “How broad is the class of normal scale mixtures?”, Ann. Statist., 65 (1978), 1159–1164 | DOI | MR

[8] C. Eisenhart, “On the transition of “Student's” $z$ to “Student's” $t$”, Amer. Statist., 33 (1979), 6–10 | DOI | MR

[9] W. Feller, An introduction to probability theory and its applications, Vol. II, John Wiley Sons Inc., New York–London–Sydney, 1966 | MR | Zbl

[10] R. A. Fisher, “Applications of “Student's” distribution”, Metron, 5 (1925), 90–104 | Zbl

[11] T. Gneiting, “Normal scale mixtures and dual probability densities”, J. Statist. Comput. Simul., 59 (1997), 375–384 | DOI | Zbl

[12] H. Hotelling, “The behavior of some standard statistical tests under nonstandard conditions”, Proc. Fourth Berkeley Symp. Math. Statist. Probab., 1 (1961), 319–360 | MR

[13] D. Kelker, “Infinite divisibility and variance mixtures of the normal distribution”, Ann. Math. Statist., 42 (1971), 802–808 | DOI | MR | Zbl

[14] J. Landerman, “The distribution of “Student's” ratio for samples of two items drawn from non-normal universes”, Ann. Math. Statist., 10 (1939), 376–379 | DOI | MR

[15] A. F. S. Lee, J. Gurland, “One-sample t-test when sampling from a mixture of normal distributions”, Ann. Statist., 5 (1977), 803–807 | DOI | MR | Zbl

[16] A. V. Makshanov, O. V. Shalaevsky, “Some problems of asymptotic theory of distributions”, Zap. Nauchn. Semin. LOMI, 74, 1977, 118–138 | MR | Zbl

[17] E. S. Pearson, “The distribution of frequency constants in small samples from nonnormal symmetrical and skew populations”, Biometrika, 21 (1929), 259–286 | Zbl

[18] P. Prescott, “A simple alternative to Student's $t$”, Appl. Statist., 24 (1975), 210–217 | DOI | MR

[19] P. R. Rieder, “On the distribution of the ratio of mean to standard deviation in small samples from nonnormal universes”, Biometrika, 21 (1929), 124–143

[20] P. R. Rider, “On small samples from certain nonnormal universe”, Ann. Math. Statist., 2 (1931), 48–65 | DOI | Zbl

[21] P. R. Rietz, “On the distribution of the “Student” ratio for small samples from certain nonnormal populations”, Ann. Math. Statist., 10 (1939), 265–274 | DOI | MR

[22] “Student”, “The probable error of a mean”, Biometrika, 6 (1908), 1–25 | DOI

[23] G. J. Székely, Paradoxes in Probability Theory and Mathematical Statistics, Reidel, Dordrecht, 1986 | MR | Zbl

[24] J. W. Tukey, D. H. McLaughin, “Less vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization,I”, Sankhya, Ser. A, 25 (1963), 331–352 | MR | Zbl