Large Deviations for Quasi-arithmetically Self-normalized Random Variables

We introduce a family of convex (concave) functions called sup (inf) of powers, which are used as generator functions for a special type of quasi-arithmetic means. Using these means we generalize the large deviation result that was obtained in the homogeneous case by Shao [14] on selfnormalized statistics. Furthermore, in the homogenous case, we derive the Bahadur exact slope for tests using self-normalized statistics. Résumé. Nous introduisons une famille de fonctions convexes (concaves) appelées sup (inf) de puissances, qui sont employées comme fonctions génératrices pour un type spécial de moyennes quasiarithmétiques. À l’aide de ces moyennes, nous généralisons le résultat que Shao [14] a obtenu dans le cas homogène sur les statistiques autonormalisées. De plus, dans le cas homogène, nous obtenons la pente exacte de Bahadur pour les tests utilisant les statistiques autonormalisées. 1991 Mathematics Subject Classification. 60F10, 62F05. The dates will be set by the publisher. Introduction The use of self-normalization in statistics dates back to Student’s t-distribution Tn := √ n Xn −m Sn where Xn := 1 n ∑n i=1Xi is the empirical mean and S 2 n := 1 n−1 ∑n i=1 ( Xi −Xn )2 is the empirical unbiaised variance of a Gaussian sample X1, . . . , Xn of mean m and unknown variance. This statistic Tn is widely used to test m = m0 against alternative and it is well known that Tn has normal limit distribution when n → ∞ (even when the variables Xi are i.i.d. non Gaussian). Shao [14] proved the associated large deviations results for i.i.d. random variables Xi that are self-normalized by homogenous means, i.e. an asymptotic estimate for P ( Xn Mp,n ≥ x ) 1 n (1)