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Fisher’s Cumulant Tests 167
normal order statistics of size n (see Table 11.6(a)). 17,18 Small values of W are evidence
of departure from normality.
Using the data in Table 11.1, a W statistic of 0.966 is obtained. This is greater
than the critical value of the W statistic at the 5 % significance level for n = 40
(see Table 11.6(b)). 17,18 Hence we have insufficient evidence to reject the null hypoth-
esis that the data did not originate from a normal distribution. For this computation,
the percentage points of the W statistics and constants, a i,n , can be obtained from Table
11.6(a).

11.6 FISHER’S CUMULANT TESTS

Fisher’s Cumulant tests are GOF tests based on the standardized third and
fourth moments of the distribution. Such tests are founded on the recognition
that deviations from normality can be characterized by the standardized third and
fourth moments of the distribution, known as the skewness and kurtosis, respectively.
Roughly speaking, skewness measure the symmetry of the distribution about the
meanandkurtosismeasurestherelativepeakednessorflatnessofthedistribution.The
normal distribution is symmetrical about the mean, and median and mode coincide
with the mean; it is said to have zero skewness. Distributions with an asymmetric tail
extending towards higher values are said to possess positive skewness (or right-handed
skew) and distributions with an asymmetric tail extending towards lower values are
said to possess negative skewness (or left-handed skew).
Kurtosischaracterizestherelativepeakednessofadistributioncomparedtoanormal
distribution. A normal distribution having a kurtosis of 0 is described as mesokurtic.
A leptokurtic distribution with positive kurtosis has a relatively more peaked shape
than the normal distribution. A platykurtic distribution with negative kurtosis has a
relatively flatter shape compared to the normal distribution. Kurtosis is relevant only
only for symmetrical distributions.
Apart from a characterization of skewness and kurtosis using the standardized
moments, these distribution characteristics can also be described in terms of the cu-
mulants of the distribution. The test statistics for the corresponding GOF tests have
been described in terms of the following first four sample cumulants which are also
called the Fisher’s K-statistics. Here Fisher’s cumulant GOF tests are described di-
rectly in terms of the sample skewness and sample kurtosis whose distribution under
normality assumption can be found in the book by Pearson and Hartley. 18
In Fisher’s cumulant GOF test for normality, the test statistic for assessing the skew-
ness of the distribution is defined as the sample skewness coefficient,

√ n 3
n (x i − x)
i=1 .
2
( n (x i − ¯x) ) 3/2
ˆ γ 1 =
i=1
A simple GOF test of normality would reject large values of | ˆ 1 |. However, since
γ
this test is targeted specifically at the skewness of distributions, it will have poor
power against any alternative distributions with γ 1 = 0 (zero skewness) which are
nonnormal.