| Summary: | 碩士 === 國立臺北大學 === 統計學系 === 91 === The multivariate normality assumption are often required when using multivariate methods. However, it is rarely examined since simple and powerful test procedures are not available. The W statistic proposed by Shapiro and Wilk (1965) is a powerful procedure for detecting departures from univariate normality. Many existing tests then generate the W statistic into the multivariate version such as the W-MA1 statistic proposed by Malkovich and Afifi (1973),the W-MA2 statistic proposed by Fatttorini (1986). Inaddition, using a transformation of the W statistic, Royston (1983) proposed an approximate test for detecting the multivariate normality. The purpose of this paper is to propose a new test procedure
W-MA3 that mimics the test W-MA1 and W-MA2.And intends to improves the drawback that the accuracy of W-MA1 and W-MA2 depend on the sample size and dimension of the sample. Owing to the unknow exact / approximate distribution of these test statistics, the Monte Carlo simulations are used to generate the percentage point of tests. The power under 9 alternative distributions and significance level of W-MA1 - W-MA3 and Royston test are also examined using simulations. All tests preserve the significance level. The W-MA3 test in general has descent power under most alternative.The W-MA1 test has least power except for the t alternative. The performance of the
W-MA2 test lies between the W-MA1 and W-MA3. The powers for the Royston test are good under most alternative, but it retains a drawback that it does not have the power in detecting the non-normality when the marginal distribution is normal.
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