Skewness refers to the symmetry of a distribution and Kurtosis refers to the flatness or ‘peakedness’ of a distribution.
This alternative space includes mixture of uniform distributions, mixture of t-distributions and the distributions used in the literature.
First and second order departures from normality depend on the skewness and kurtosis of the distribution, we have used 72 alternatives with wider ranges of these parameters.
At the same time, the space should be large enough to provide a good approximation to the full space of alternatives–failing that, it should be large enough to approximate the distributions conventionally used in simulations studies. Since we plan to use numerical methods, we must narrow this space down to something sufficiently small to permit exploration by numerical methods. Another problem is that the alternative space is infinite dimensional.
This method reduces the calculation burden in terms of computing and estimating the Neyman-Pearson test against each alternative distribution for the benchmark as proposed in. Maximum deviations of all selected tests from the benchmark is computed and the test with minimum deviation is ranked as best. The best test’s performance against each alternative distribution provides us the benchmark for comparison of normality tests by using the max-min criterion. This study rests on the finding that one normality test is optimal against one alternative and another for another alternative distribution. Comparison of normality tests via simulations are bound to give ambiguous results, since these statistics critically depend upon the alternative distribution which cannot be specified. Thus, one normality statistic may perform well for one alternative distribution and another for another alternative non-normal distribution. Different characteristics of normal distribution are exploited while developing normality statistics consequently the power of normality tests varies, depending upon the nature of non-normality. With the development of several normality tests over the decades, power comparison of these statistics has been given the due consideration in literature in search of the best test thus helping the researchers in the choice of suitable normality test. Given the importance of the subject, literature has produced a plethora of goodness-of-fit tests to detect departures from normality. , find only 5.5 percent of the 693 real data distributions close to normality while considering skewness and kurtosis together. Diagnostic tests for normality are important as Blanca et al. In short, normality assumption of the given data is the key to validate the inferences made from regression models and other statistical procedures. The experimental data sets generated in clinical chemistry for the construction of population reference ranges require the assumption of normality. Statistical inference from regression models applied to time series, categorical and count data depends crucially on the assumption of normal errors.
In both cross-sectional and time series data, assuming the data normality without testing may affect the accuracy of the econometric inference. Normality of the data is the underlying distributional assumption of multitude of statistical procedures and estimation techniques. The proposed min-max method produces similar results in comparison with the benchmark based on Neyman-Pearson tests but at a low computational cost. An extensive simulation study is conducted to evaluate the selected normality tests using the proposed methodology. The proposed min-max approach reduces the calculation cost in terms of computing and estimating the Neyman-Pearson tests against each alternative distribution. However, the computational cost of this benchmark is significantly high, therefore, this study proposes an alternative approach for computing the benchmark. Thus, an invariant benchmark is proposed in the recent normality literature by computing Neyman-Pearson tests against each alternative distribution. A test which is optimal against a certain type of alternatives may perform poorly against other alternative distributions. Comparison of normality tests based on absolute or average powers are bound to give ambiguous results, since these statistics critically depend upon the alternative distribution which cannot be specified.