The name is absent

We can write the cdf F and the quantile function F~¹ corresponding to a skew
density f of an X distribution using the cdf G and the quantile function G~¹ of the
symmetric density. We have

for X < 0,
for X > 0.

for the cdf and

f ⅛G^¹ [(1 + κ²)⅞] for x < 0,
F~¹(x) = < ^{L        2J}            ~

ɪ-κG~¹ [(1 + κ^-2)++ for x > 0,

for the quantile function. Moreover, the moments of order r, r = 1, 2,..., can be written
as

/.'^r÷ ' 4- (~ ŋ

E(X^r) = 2E⁺(X^r)-------^K~^(r+1) ,

^κ + ∑

where                                    _oo

E⁺(X^r) = / x^rg(x)dx

Jo

is the r-th moment of the symmetric distribution truncated to positive values. Clearly
E⁺(X^r) assumes the value of the r-th moment of the symmetric distribution divided
by 2 when r is even.

On the basis of these formulations, we present the main characteristics of the dis-
tributions considered (for details see, among others, Ayebo and Kozubowski, 2003;
Lambert and Laurent, 2001; Palmitesta and Provasi, 2004).

Skew Student-t. A random variable X has a skew Student t distribution (SST) if the
parameters к > 0 and v > 2 exist such that the pdf of X is

where Γ(∙) indicates the gamma fucntion. If к = 1, the distribution is symmetric on
zero and has zero mean and unit variance. The cdf of X is given by

for x < 0,

for x > 0,

Γ(4±l) /     2∖- C + l)∕2

t+)= ЛгД ^ι+- ^dw
J-₀₀ √≡Γ(+ V V J

is the cdf of the unsealed Student t with υ degrees of freedom.

The moments of order r, r = 1, 2,..., of the SST distribution are:

and exist when v > r.

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