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4: Stefan A. Robila
Kurtosis is highly sensitive to outliers. A small number of outliers can significantly affect its value. For example, Fig. 4.2a shows the histogram of a Gaussian
random variable with zero mean and standard deviation one generated from
100,000 observations. Figure 4.2b shows the histogram of the same random
variable perturbed by the addition of 500 observations generated by a Gaussian random variable with mean value as five and standard deviation as one.
The perturbation is almost unnoticeable in the histogram even when enlarged
(Fig. 4.2d). However, this small perturbation was sufficient to change the kurtosis value from close to zero (0.0022) (before) to (2.5639) (after) showing its
sensitivity to outliers.
The kurtosis for super-Gaussian variables can have large positive values (in
theory up to infinity) whereas it is limited to -2 for sub-Gaussian variables
(Hyvarinen et al. 2001). There is a possibility for a random variable to have
the kurtosis equal to zero without being Gaussian. However, this occurrence is
rare.
7000r---~--~--~-~
7000 r----_--~--_-____,
6000
6000
5000
kurtosis=0.0022
5000
kurtosis=2.5639
4000
4000
3000
3000
2000
2000
1000
OL-------~-~~------~
I
0L-______ ~_~~ ______ ~
1000
-10
-5
o
5
10
-10
-5
o
5
10
a
b
2 0 0 r - - - - - - - - - - - - - - - ,
150r---~--~--~-~
150 \
100
~~
°2L--~~3------4------5------6
/
100
6
c
d
Fig.4.2. Example of kurtosis instability. a Gaussian distributed random variable (mean =
0, standard deviation = 1 and number of observations = 100,000) b A perturbation of
500 observations is added on the right tail c, d Area enlarged of the two graphs (a and b
respectively) showing that the perturbation is almost unnoticeable in the histogram