Sin t

1
Influence Function Approach to Sensitiveness of
Kurtosis Indexes
Francesco Campobasso, Dipartimento di Scienze
Statistiche Carlo Cecchi, Università degli Studi
di Bari, Italy, e-mail fracampo_at_dss.uniba.it
Introduction
Fiori and Zenga (2005) recently analysed the
meaning of the kurtosis index introduced by K.
Pearson, starting from a Faleschinis
pioneeristic work (1948) and arriving to identify
its answer with respect to the frequencies of
each modality. The authors achieved the same
results as those achieved by Faleschini, using
the more modern methodology of the influence
function suggested by Hampel (1974) and utilized
in this context by Ruppert (1987).The aim of this
paper is to go through these results and to
compare the kurtosis index attributed to Gini
with Pearsons one.

The kurtosis index attributed to Gini is given by
Gs2/d2, where s is the standard deviation and d
is the mean deviation of the distribution. It is
well known that always G1, that for the normal
distribution Gp/2 and that values of G smaller
(larger) than p/2 indicate platikurtosis
(leptokurtosis). With reference to the originary
Faleschinis procedure, we study the partial
derivatives of the kurtosis index with respect to
the frequencies of each modality. As the partial
derivatives of s2 and d with respect to the
frequency nr (r1,2,...,s ) are
and
, where
is the size and is the
arithmetic mean of the data, it follows that
the partial derivative of the index G with
respect to the frequency nr is
By substituting
, setting the derivative equal to
zero and dividing for the nonnegative
quantity G, we obtain the equation
, whose roots are
. In the case of the
standard normal distribution, as Gp/2, the range
of the derivative is divided in the following
sections (-8, -2,009),
(-2,009, -0,498), (-0,498, 0,498), (0,498,
2,009), (2,009, 8).
Lets consider - only for simplicity - a
variable T having a cumulative distribution
function F(t), such that E(T)0 and var(T)
s2. If we contaminate the variable T at the point
x through a quantity e (0ltelt1), the cumulative
distribution function of the new variable is
where H(t) equals 0 for t lt x and 1 for t x.
Moreover its mean is
, its variance is
as Fiori and
Zenga (2005) derived by performing some
computations, and its mean deviation is As the
first integral can be split in two parts (from -8
to ex and from ex to 8), while the second one
equals , the mean deviation can be
written as Noting that the integral of the
constant ex from -8 to 8 equals ex and the
integral of t from -8 to 8 equals zero (as T is
expressed in terms of deviation from its mean),
the previous expression becomes where F1(t)
is the incomplete first moment of T. At this
point we can use the following relationship
between F(t) and F1(t)
in order to derive

It is just the case
to note that The obtained expressions of
and enables to derive the influence function of
G Preliminarily we verify that
and
, where
equals 1 for x lt 0 and 0 for x 0 then we
find, by means of de LHospitals rule,
that In the case of symmetrical variables,
in which F(0)1/2, after considering the
standardized value , the influence function
becomes At last, by setting the influence
function equal to zero and dividing it for the
nonnegative quantity G, we obtain the
equation which is the same as the one obtained
by means of Faleschinis procedure.
The influence function in the continuous case
The sensitiveness of kurtosis index attributed to
Gini
Final remarks
Fig.1 shows the behaving of the derivatives of G
and ?1 with respect to zr in the case of the
standard normal distribution. The comparison of
the two derivatives shows that G is generally
less sensitive to contaminations than ?1. Indeed,
when attention is focused on the middle part of
the distribution, the interval where a
contamination determines an increase of kurtosis
is narrower for the G index on the other hand,
when attention is focused on the tails of the
distribution, the intervals where a contamination
determines an increase of kurtosis are larger for
the G index . The conjoint use of the two indexes
enables a better evaluation of the
nonnormality. Figure 1 Derivatives of G and ?1
for the standard normal distribution

REFERENCES
Faleschini L. (1948) Su alcune proprietà dei
momenti impiegati nello studio della variabilità,
asimmetria e curtosi, Statistica, 8,
503-513. Fiori A.M., Zenga M. (2005)
Linterpretazione della curtosi e la sua curva di
influenza in unintuizione di L. Faleschini,
Statistica, anno LXV, 2005, 2, 135-144. Hampel
F.R. (1974) The influence curve and its role in
robust estimation, Journal of the American
Statistical Association, 69, 383-393. Ruppert D.
(1987) What is kurtosis? An influence function
approach, The American Statistician, 41, 1, 1-5.