Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University20827Original ArticleThe use of hermite special functions for investigation of power properties of grubbs statisticsShiryaevaLudmila K(Ph. D. (Phys. &amp; Math.)), Associate Professor, Dept. of Mathematic Statistics and Econometricsshiryeva_lk@mail.ruSamara State Economic University1512201216413114518022020Copyright © 2012, Samara State Technical University2012We consider a normal sample with a single upper outlier. A distribution of studentized form of outlier’s deviation from the sample mean is obtained. This distribution uses Hermite special functions with negative integer-valued index. The integral relationships for David’s power measures of Grubbs criteria are obtained. We discuss the case, when Grubbs statistic is the likelihood-ratio statistic. We ﬁnd the maximal deviation of power function for Grubbs criteria from the probability that the contaminant is the outlier and it is identiﬁed as discordant. We receive the region of critical values of Grubbs statistic, where the second power measure of David equals to the third and forth power measures of David. We make calculations of power function for Grubbs criteria in the case of normal samples with a single upper outlier with the right shift. The results of calculations are similar to the theoretically expected facts.outlierGrubbs statisticsHermite special functionDavid’s power measures for Grubbs criterionnormal distribution lawвыброскритерий Граббса проверки на один выбросфункция Эрмитамеры мощности Дэйвида для критерия Граббсанормальный закон распределенияDavid H. A. Order Statistics: 2nd ed. / Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons, 1981. xiii+360 pp.Grubbs F. E. Sample criteria for testing outlying observations // Ann. Math. Statistics, 1950. Vol. 21, no. 1. Pp. 27–58.Ширяева Л. К. Вычисление мер мощности критерия Граббса проверки на один выброс // Сиб. журн. индустр. матем., 2010. Т. 13, № 4. С. 141–154.Zhang J., Keming Y. The null distribution of the likelihood-ratio test for one or two outliers in a normal sample // Test, 2006. Vol. 15, no. 1. Pp. 141–150.Barnett V., Lewis T. Outliers in statistical data: 3nd ed. / Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Chichester: John Wiley & Sons, 1994. xviii+584 pp.Айвазян С. А., Мхитарян В. С. Теория вероятностей и прикладная статистика. Т. 1. М.: Юнити-Дана, 2001. 656 с.Лебедев Н. Н. Специальные функции и их приложения. М.: Физ.-мат. лит., 1963. 358 с.