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Fisher’s Cumulant Tests 169
Table 11.7 Critical values of moment statistics for Fisher’s cumulant GOF test. 18
Sample skewness (γ 1 ) Sample skewness (γ 2 )
α = 5% α = 10 % α = 5% α = 10 %
Sample size, n Upper* Upper* Upper (Lower) Upper (Lower)

20 0.940 0.772 1.68 (−1.27) 1.18 (−1.17)
25 0.866 0.711 -- --
30 0.806 0.662 1.57 (−1.11) 1.12 (−1.02)
35 0.756 0.621 -- --
40 0.714 0.588 1.46 (−1.01) 1.06 (0.93)
45 0.679 0.559 -- --
50 0.647 0.534 1.36 (−0.94) 1.00 (−0.85)
100 0.470 0.390 1.03 (−0.73) 0.77 (−0.65)

*Lower limits here are equivalent to the corresponding upper limits with negative sign.

The test statistic for assessing the kurtosis of the distribution is defined by the sample
kurtosis,
n 4
n (x i − ¯x)

i=1 − 3.
2
ˆ γ 2 =
( n (x i − ¯x) ) 2
i=1
Percentage points for different critical values of ˆ 1 and ˆγ 2 for these GOF tests can
γ
be found in Table 11.7. 18
For large samples (n > 100), the following test statistic can be used instead of the
sample skewness:
ˆ γ 1
Z 1 = √ .
6/n
Similarly, for large sample size, the following test statistic can be used in place of the
sample kurtosis:
ˆ γ 2 − 3
Z 2 = √ .
24/n
Both of these statistics can be approximated by a standard normal distribution.
Consequently, there is a combined test which simultaneously takes into account both
the skewness and kurtosis of the distribution. This is given by
2
2
2
X = Z + Z ,
1 2
2
which is approximately χ distributed with 2 degrees of freedom.
The skewness and kurtosis measures computed for the data in Table 11.1 are −0.020
and −0.933, respectively. The critical values at 5% significance level for a sample
size of n = 40 can be obtained from Table 11.7. Two-sided ˆ 1 and ˆγ 2 tests both yield
γ
p-values that are greater than 0.05. Hence, the null hypothesis that the data comes
from a normal distribution cannot be rejected at the 5 % level of significance.
2
2
The combined X statistics is 1.452. This is lower than the critical χ value at the 5 %
significance level, hence the null hypothesis cannot be rejected at the 5 % significance