Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, Iran a Department of Mathematics, Iran University of Science and.

Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, Irana

Department of Mathematics, Iran University of Science and Technology, Tehran, Iranb

Department of Mathematics, Islamic Azad University, Gahram, Iranc

Introduction

where :

The usual kurtosis measure is

What is kurtosis ?

does not sort the distributions based on the height

Height

We compute for some distributions,

Uniform, Normal, Logistic, Laplace, and Cauchy for the above distributions the kurtosis Measures, ,are 1.8, 3, 4.2, 6,

and respectively.

What is kurtosis ?

Height

As the height of distributions increases,

the order of distributions will be Logistic, Cauchy, Normal, Laplace, and

Uniform respectively.

What is kurtosis ?

Height

Disadvantages of

1- It’s infinite for heavy tail distributions.

2- It doesn’t work well for some distributions such as Ali’s scale contaminated normal distributions.

where is the standard normal distribution function.

3- It can be misleading as a departure from normality.

when

But this sequence converges in distribution to the standard normal distribution as .

is not a sufficient condition for normality.

A Modified Measure of Kurtosis

Where p and q are quantile of order p and q, respectively

Properties of If f(x) > 0 for all x.

P.3. =

Proof:

since

where A(k) = And B(k) =

Lim A(k) = lim B(k) = 0 as

We show that the treatment of is as the same as

Oja (1981) says a location and scale invariant functionl T can be

named a kurtosis measure if whenever G has at leastas much Kurtosis as F according to the definition of relative kurtosis.

T is a kurtosis measure if : 1- It must be location and scale invariant i.e. T(ax+b) = T(x) for a > 0 2- It must preserve one of the orderings. Ordering << were defined in such a way that F<< G means, in some location and scale free, that G has at least as much mass in the center and tails as F i.e. if F<sG then T(F) <= T(G)

Properties of kurtosis measure

Van Zwet (1964) introduced for the class of symmetric distributions an ordering <s

defined by F <s G iff is convex for x > mf. Where mf is the point of symmetric of F. The distributions are ordered by the s ordering.

SIF < 0 iff .742 < x < 2.334 SIF > 0 iff x < .742 or x > 2.334

center

flank

tail

6 = β2(Laplace) > β2(Logistic) = 4.2

By comparing SIF of two distributions the centers of two distributions are approximately equal.

But contamination at the tails of Laplace is more than Logistic.

Contamination at the center has far less influence than that in the extreme values.

We compute SIF for Logistic and Laplace to compare the concentration probability mass in their center, flank and tails.

Let F and G be distribution functions that are symmetric and define

We consider

G =

Shape Parameter of sharply peakedness

The values of are equal to 6 for all values of

Discussion:We discuss about the properties of the usual kurtosis, then we introduce a modified measure of kurtosis and we consider the properties of this kurtosis measure.We show that this kurtosis measure is robust and its treatment is as the same as the usual kurtosis measure. Furthermore for departure from normality the shape of the parameter must be considered, so by noting the sharply peaked parameter the inference of distribution of population will be more correctly.

For < .798 the distribution is shorter than standardized normal distribution. By increasing the distributions willbe more sharper.

References: