Adjusting the tests for skewness and kurtosis for distributional misspecifications

The standard root b(1) test is widely used for testing skewness. However, several studies have demonstrated that this test is not reliable for discriminating between symmetric and asymmetric distributions in the presence of excess kurtosis. The main reason for the failure of the standard test is that the expression of its variance, generally used in practice, is derived under the assumption of no excess kurtosis. In this article, we theoretically derive adjustment to the test under the framework of Rao's score (or the Lagrange multiplier) test principle. Our adjusted test automatically correct the variance formula and does not lead to over- or under-rejection of the correct null hypothesis of no-skewness. In a similar way, we also suggest an adjusted test for kurtosis in the presence of asymmetry. These tests are then applied to both simulated and real data. The finite sample performances of the adjusted tests are far superior compared to those of their unadjusted counterparts. For a proper comparison, we also consider Monte Carlo tests in our study and find those to be quite effective in testing for skewness and excess kurtosis in the presence of possible distributional misspecifications.