A New Goodness of Fit Test for Normality with Mean and Variance Unknown.

Abstract : A new technique for calculating known goodness of fit statistics for the Normal distribution is investigated. Samples are generated for a Normal (0, 1) distribution. The means of these samples are calculated and the samples are doubled by reflecting sample data points about the individual sample means. This reflection of data points about the individual sample means. This reflection of data points about the mean is the new technique for generating modified statistics. After the sample is doubled, critical values are calculated for these modified Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises statistics. Critical values are for the original sample sizes. An extensive power study is done to test the power of the three new statistics' critical values versus the power for the same three statistics, calculated without reflection. Powers of the new statistics are asymptotically slightly higher than the powers of their non-reflected counterparts, when the distribution tested is also symmetrical. The powers of new statistics are substantially lower when the distribution tested is non-symmetrical. The powers are substantially higher for the modified statistics when the continuous Uniform distribution is tested. Complete tables of critical values for sample sizes n = 3 through n = 60 are included for the modified Kolmogorov-Smirnov and Anderson-Darling statistics. (Author)