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The characterization results in probability and statistics are better known for their structural aesthetics and mathematical challenge than for their potential in the development of statistical methodology. This paper shows independence characterizations that can motivate and yield tests of normality for detecting a variety of restricted skewness and kurtosis alternatives. First we examine the Z test (Lin and Mudholkar, Biometrika 67 (1980) 455), now labelled the Z 2 test, which is tied to the independence of the mean and the variance of normal samples and is sensitive to skew alternatives. In a similar manner, we use the characteristic independence of the mean and the third central moment of normal samples to construct a new statistic denoted by Z 3 , which is seen to be well-suited for detecting nonnormal kurtosis. The Z 2 and Z 3 statistics are shown to be asymptotically normal and mutually independent. It is empirically demonstrated that the two statistics are uncorrelated for samples of moderate size, and that the tests based upon them may be combined effectively to test for normality against restricted or unrestricted skewness and kurtosis alternatives. The power functions of the tests are compared to those of the famous tests such as tests based on sample skewness and sample kurtosis, tests due to Shapiro and Wilk, D'Agostino and others. |
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