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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.