Title: Truncated Cramér-von Mises test of normality,
Abstract: A new test of normality based on a standardised empirical process is
introduced in this article.
The first step is to introduce a Cramér-von Mises type statistic with
weights equal to the inverse of the standard normal density function supported
on a symmetric interval $[-a_n,a_n]$ depending on the sample size $n.$ The
sequence of end points $a_n$ tends to infinity, and is chosen so that the
statistic goes to infinity at the speed of $\ln \ln n.$ After substracting the
mean, a suitable test statistic is obtained, with the same asymptotic law as
the well-known Shapiro-Wilk statistic. The performance of the new test is
described and compared with three other well-known tests of normality, namely,
Shapiro-Wilk, Anderson-Darling and that of del Barrio-Matrán, Cuesta
Albertos, and Rodr\'{\i}guez Rodr\'{\i}guez, by means of power calculations
under many alternative hypotheses.