# t-Tests with Models Close to the Normal Distribution

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## Abstract

The *t*-distribution is a very usual distribution for several test statistics because a normal distribution is frequently assumed as underlying model. Even in some tests based on robust statistics, such as the test based on the sample trimmed mean, a *t*-distribution is used as distribution for the standardized sample trimmed mean if the underlying model is normal. Nevertheless, it is necessary to have a deeper understanding of the behaviour of these kind of tests and the computations of their key elements, such as the *p*-value and the critical value, with small samples, when the underlying model is close but different from the normal distribution. In this paper, we obtain good analytic approximations, with small samples, for the *p*-value and the critical value of a *t*-test (i.e., a test with a *t*-distribution for the test statistic under a normal model), studying its behaviour when the underlying distribution is close but different from the normal model. We conclude the paper with a discussion on some robustness properties of *t*-tests.

## Keywords and phrases

Robustness in hypotheses testing von Mises expansion tail area influence function saddlepoint approximation robustness of*t*-tests

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## References

- 1.Benjamini, Y. (1983). Is the
*t*test really conservative when the parent distribution is long-tailed?,*Journal of the American Statistical Association*,**78**, 645–654.zbMATHCrossRefMathSciNetGoogle Scholar - 2.Chen, H., and Loh, W. Y. (1990). Uniform robustness against nonnormality of the
*t*and*F*tests,*Communications in Statistics—Theory and Methods*,**19**, 3707–3723.zbMATHCrossRefMathSciNetGoogle Scholar - 3.Cressie, N. (1980). Relaxing assumptions in the one sample
*t*-test,*Australian Journal of Statistics*,**22**, 143–153.zbMATHMathSciNetCrossRefGoogle Scholar - 4.Daniels, H. E. (1983). Saddlepoint approximations for estimating equations,
*Biometrika*,**70**, 89–96.zbMATHCrossRefMathSciNetGoogle Scholar - 5.Field, C. A., and Ronchetti, E. (1985). A tail area influence function and its application to testing,
*Communications in Statistics—Theory and Methods*,**14**, 19–41.MathSciNetGoogle Scholar - 6.Filippova, A. A. (1961). Mises’ theorem on the asymptotic behaviour of functionals of empirical distribution functions and its statistical applications,
*Theory of Probability and Its Applications*,**7**, 24–57.CrossRefGoogle Scholar - 7.García-Pérez, A. (2003). Von Mises’ approximation of the critical value of a test,
*Test*,**12**, 385–411.zbMATHCrossRefMathSciNetGoogle Scholar - 8.García-Pérez, A. (2004). Chi-square tests under models close to the normal distribution,
*Technical Report*, Departamento de Estadística, Investigación Operativa y Cálculo Numérico, UNED, Madrid.Google Scholar - 9.Jensen, J. L. (1995).
*Saddlepoint Approximations*, Clarendon Press, New York.Google Scholar - 10.Lee, A. F. S., and Gurland, J. (1977). One-sample
*t*-test when sampling from a mixture of normal distributions,*The Annals of Statistics*,**5**, 803–807.zbMATHMathSciNetGoogle Scholar - 11.Loh, W. Y. (1984). Bounds on AREs for restricted classes of distributions defined via tail-orderings,
*The Annals of Statistics*,**12**, 685–701.zbMATHMathSciNetGoogle Scholar - 12.Lugannani, R., and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables,
*Advances in Applied Probability*,**12**, 475–490.zbMATHCrossRefMathSciNetGoogle Scholar - 13.Patel, K. R., Mudholkar, G. S., and Fernando, J. L. I. (1988). Student’s
*t*approximations for three simple robust estimators,*Journal of the American Statistical Association*,**83**, 1203–1210.CrossRefMathSciNetGoogle Scholar - 14.Reeds, J. A. (1976).
*On the Definitions of Von Mises’ Functionals*, Ph.D. Thesis, Department of Statistics, Harvard University, Cambridge, MA.Google Scholar - 15.Sawilowsky, S. S., and Blair, R. C. (1992). A more realistic look at the robustness and type II error properties of the
*t*test to departures from population normality,*Psychological Bulletin*,**111**, 352–360.CrossRefGoogle Scholar - 16.Serfling, R. J. (1980).
*Approximation Theorems of Mathematical Statistics*, John Wiley_& Sons, New York.zbMATHGoogle Scholar - 17.Staudte, R. G., and Sheather, S. J. (1990).
*Robust Estimation and Testing*, John Wiley_& Sons, New York.zbMATHGoogle Scholar - 18.Tukey, J. W., and McLaughlin, D. H. (1963). Less vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization 1,
*Sankhyā, Series A*,**25**, 331–352.zbMATHMathSciNetGoogle Scholar - 19.Wilcox, R. R. (1997).
*Introduction to Robust Estimation and Hypothesis Testing*, Academic Press, San Diego.zbMATHGoogle Scholar