To a given characteristic function [lambda] over P, we can assign an n x m [Z.sub.2]-matrix [LAMBDA] by ordering the facets and choosing a basis {[e.sub.1], ..., [e.sub.n]} of [Z.sup.n.sub.2] by

In each single stage game r, the characteristic function V(S),S [subset] N is defined by V(S) = Val[[GAMMA].sub.SN\S], where Val[[GAMMA].sub.SN\S] is a value of zero-sum game played between coalition S acting as player 1 and coalition N\S acting as player 2.

To find an approximating density function of temperature, inversion formula will be applied to the characteristic function of the temperature defined in (15).

We compare the performance of the introduced U-statistic for the tail index with the well-known methods, including maximum likelihood, empirical characteristic function, sample quantile, and that introduced in Mohammadi et al.

If the parameter [beta] = 0, X is called symmetric a stable distribution (S[alpha]S), its characteristic function can be expressed as (2), and when [alpha] = 2, which is the same as the characteristic function of Gaussian distribution,

However, the estimation of the symmetry parameter [beta] and location parameter [mu] can be calculated with the estimated value of [alpha] and [gamma], resulting in cumulative error propagation of such [beta] and [mu] Meanwhile, because the characteristic function of a stable distribution is intermittent in x = 1, the estimation error is particularly serious in [beta] [not equal to] 0 and [alpha] [right arrow] 1.

The statistical properties of this new class, such as mean, variance, standard deviation, mean deviation, kurtosis, skewness, moment generating function, characteristic function and graphical analysis, are derived.