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        162              2:57  Goodness-of-Fit Tests for Normality
                                 2
        Table 11.5 Modifications of A and W statistics for mean, μ, and variance, σ , unknown*
                                       2
                                                                      2
        and the corresponding critical values [9].
                                                                Significance level (α)
        EDF based
        Statistic                   Scaling                   0.05            0.10
                                                2
                              2
        A 2                  A (1.0 + 0.75/n + 2.25/n )       0.752           0.631
                                   2
        W  2                     W (1.0 + 0.5/n)              0.126           0.104
        *Mean, μ, estimated from ¯x, and standard deviation, σ, estimated from
           2         2
          s =   (x i − ¯x) /(n − 1)
              i
        11.4.3 Anderson--Darling
        Another popular EDF based GOF test for normality is based on the Anderson--Darling
        test statistic. Like the Cram´er-von Mises test statistic, it is based on the quadratic
        measure of deviation shown in (11.3), but with weighting function


                                −1
          ψ(x) = F 0 (x)[1 − F 0 (x)]  .


        Such a weighting function results in deviations in the tails of distributions (when
        F 0 (x) = 0 or 1) being weighted more heavily. Hence, this test is more sensitive to
        deviations in the tails of distributions. For ordered observations (x (i) )of nobservations,
                                         2
        the Anderson--Darling test statistic, A , is given by
                       n
                    1
           2
          A =−n −        (2i − 1) ln F 0 (x (i) ) + ln[1 − F 0 (x (n+1−i) )] .  (11.6)
                    n
                      i=1
                                  2
        The limiting distribution of A is given in Anderson and Darling’s original paper. 16
        As this statistic requires the use of specific distributions in calculating the critical
        values, it possesses the advantage of being a more sensitive test. However, the critical
        values have to be calculated for each distribution. This test can be used for normal,
        lognormal, exponential, Weibull, extreme value type I and logistic distributions.
          We return to the residuals data set in Table 11.1 to demonstrate the Anderson--
        Darling GOF test. As before, the hypothesis that this data set comes from a normal
        distribution is tested with the assumption that the population mean is zero and the
                                                                          2
        standard deviation unknown and estimated from data. Using equation (6), A is found
                                                 2
        to equal 0.262. This is lower than the critical A value at the 5% significance level of
                                                                        2
        2.323 shown in Table 11.4(b). 15  For completeness, modifications to the A statistic in
        order to deal with GOF tests where the population mean and standard deviation
                                                  9
        are estimated from data are given in Table 11.5. Critical values for the modified A 2
        statistics for this case are also reproduced in Table 11.5 together with the scaling for
                                      2
        a modified Cram´er--von Mises W statistic and its corresponding critical values for
        cases where the population mean and standard deviation are unknown.
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