Page 113 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
(b) For a positive number we let denote the Gamma distribution given

by

3
1
It was proved in [10] that ∈ ℐ(⊞),if ∈ (0, ] ∪ [ , ∞),whereas ∉ ℐ(⊞),

2
2
if belongs to the set

(c) For any positive number we denote by −1 the inverse Gamma-
distribution given by

It was proved in [10] that −1 ∈ ℐ(⊞) for all in (0, ∞).
(d) For positive numbers , we let , denote the Beta-distribution given
by

3
In [10] it was proved that, , ∈ ℐ(⊞), if , ≥ , or if + ≥ 2 and either
2
1
or belongs to (0, ].
2
If , ∈ (0,1], or if either or belongs to the set ℐ given in (3.1), then , ∉
ℐ(⊞).
(e) For , in (0, ∞) we let ′ , denote the Beta-distribution of second kind
given by

1 3
It was proved in [10] that′ , ∈ ℐ(⊞), if ∈ (0, ] ∪ [ , ∞), whereas ′ , ∉
2
2
ℐ(⊞), if ∈ ℐ.
1
(f) For any number in ( , ∞) we denote by Student’s -distribution given
2
by

1 1

It was proved in [10] that ∈ ℐ (⊞), if ∈ ( , 2] or if ∈ ⋃ ∈ℕ [2 + , 2 +
2
4
2].
Some of the results from [10] listed above were obtained previously in
special cases; we refer to [10] for a full bibliographical account on such partial
results.

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