## On the Bahadur slope of the Lilliefors and the Cramér-von Mises tests of normality.(English)Zbl 1153.62034

Giné, Evarist (ed.) et al., High dimensional probability. Proceedings of the fourth international conference. Many papers based on the presentations at the conference, Santa Fe, NM, USA, June 20–24, 2005. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-67-6/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 51, 196-206 (2006).
From the paper: The simplest goodness of fit testing problem is to test whether a random sample $$X_1,\dots,X_n$$ is from a particular c.d.f. $$F_0$$. The testing problem is:
$H_0 : F = F_0,\text{ versus }H_1 : F\not\equiv F_0.$
A common goodness of fit test is the Kolmogorov-Smirnov test. The Kolmogorov-Smirnov test rejects the null hypothesis for large values of the statistic
$\sup_{t\in\mathbb R}|F_n(t)-F_0(t)|,$
where $$F_n(t) =n^{-1}\sum^n_{j=1}I(X_j\leq t)$$, $$t\in\mathbb R$$, is the empirical c.d.f. Another possible test is the Cramér-von Mises test, which is significative for large values of the statistic:
$\int^\infty_{-\infty}[F_n(t)-F_0(t)]^2 \,dF_0(t).$
H. Lillefors [J. Am. Stat. Assoc. 62, 399–402 (1967)] proposed the normality test which rejects the null hypothesis for large values of the statistic
$\sup_{t\in\mathbb R}|F_n(\overline X_n+s_nt)-\Phi(t)|,\tag{*}$
where $$\Phi$$ is the c.d.f. of the standard normal distribution, $$\overline X_n := n^{-1}\sum^n_{j=1}X_j$$ and $$s^2_n:=(n-1)^{-1}\sum^n_{j=1}(X_j-\overline X_n)^2$$. This test can be used because the distribution of (*) is location and scale invariant. Here the Bahadur slope of the Lilliefors and Cramér-von Mises tests of normality is given.
For the entire collection see [Zbl 1113.60009].

### MSC:

 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 60F10 Large deviations
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