Abstract:
Normality is an essential underlying assumption of several statistical procedures. Dozens of
tests are available in the literature to test the hypothesis of the normality of data. A plethora of
simulation studies yielded different optimal tests for different alternative distributions which
cannot be specified. Normality tests are based on the different characteristics of normal
distribution. Thus, it is not possible to find an optimal test against all non-normal distribution.
This study aims to find an optimal test based on sample sizes. Monte Carlo procedures are
called in to compute the loss function and tests are ranked using the min-max criterion. We
recommend Shapiro -Wilk (SW), Shapiro-Francia (SF) & Barrio et al. (BCMR) tests for small,
medium and large sample sizes if the alternative space consists of slightly skewed distributions;
for moderately skewed alternative space, Shapiro-Wilk (SW) & Chen-Shapiro (CS) are
considered optimal for all sample sizes. In case of highly skewed alternatives, SF, CS & SW tests
are best for small sample sizes whereas for medium and large sample sizes all of the tests turn
out to be the best except for COIN and Bonett and Seier (Tw). On balance, the SW test turns
out to be the most stringent test for all sample sizes against the entire selected alternative space.
