Methods for Testing Normality of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 44

need to be calculated for each distribution. The Anderson-Darling test is an
alternative to the chi-square and Kolmogorov-Smirnov tests. It makes use of
the fact that in case of a hypothesized underlying distribution, and assuming
the data does arise from this distribution, the data can be transformed to a
uniform distribution. Thereafter, the transformed sample data can be tested for
uniformity (Shapiro, 1980).
The test-statistic ( A ) for the Anderson-Darling test to evaluate, if the data
( Y 1 < Y 2 <…< Y n ) comes from a distribution with cumulative distribution function
(CDF) F , is given as (Stephens, 1986):
A 2 =- n - S

(4)

n

21 ln k

Ç

> @

where S F Y ( ) ln (1 F Y ( )) (5)

k n+1- k

n

k 1

Note that the time series data need to be arranged in decreasing order (i.e.,
Y 1 < Y 2 <…< Y n ) before computing the test-statistic, A . The value of A thus
computed is compared with the corresponding critical value of the theoretical
distribution. This test is a one-sided test and the hypothesis that the distribution
is normal is rejected if the value of A is greater than the critical value.
Despite excellent theoretical properties of the Anderson-Darling test, it
has a serious flaw when applied to real-world time series data. The Anderson-
Darling test is severely affected by ties in the data because of poor precision
(Stephens, 1986). When a significant number of ties exist in a dataset, the
Anderson-Darling test will often reject the data as non-normal, irrespective of
how well the data fit the normal distribution.

3.2.5 Cramér-von-Mises Test

The Cramér-von-Mises test is an alternative to the Kolmogorov-Smirnov test
and the Anderson-Darling test. Let x 1 , x 2 , …, x n be the observed values of a
hydrologic time series in increasing order. The Cramér-von-Mises statistic is
computed as (Stephens, 1986):

2

n

21 i 1

È Ø

Ç

W 2 =

Fx ()

(6)

É Ù

i

Ê Ú

2 n 12 n

i 1

where F ( x ) = distribution function of x and n = sample size of the time series.
If the value of the test-statistic is larger than the corresponding critical value,
the hypothesis that the data come from the distribution F is rejected.

3.2.6 Shapiro-Wilk Test

The Shapiro-Wilk (S-W) test is one of the most powerful and omnibus normality
test (Shapiro, 1980; Gilbert, 1987; USEPA, 2006). This test is similar to
computing a correlation between the quantiles of the standard normal