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Table 1 The conditions on the parameters and the support regions of GLD (p. 39, Karian and Dudewicz (2010))

From: Families of distributions arising from the quantile of generalized lambda distribution

Region

λ 1

λ 2

λ 3

λ 4

Q(0)

Q(1)

1

all

<0

< − 1

>1

−∞

λ 1 + (1/λ 2)

5

all

<0

\( \left\{\begin{array}{c}-1<{\lambda}_3<0,\kern0.5em {\lambda}_4>1\\ {}\frac{{\left(1-{\lambda}_3\right)}^{1-{\lambda}_3}{\left({\lambda}_4-1\right)}^{\lambda_4-1}}{{\left({\lambda}_4-{\lambda}_3\right)}^{\lambda_4-{\lambda}_3}}<\frac{-{\lambda}_3}{\lambda_4}\end{array}\right. \)

−∞

λ 1 + (1/λ 2)

2

all

<0

>1

< − 1

λ 1 − (1/λ 2)

6

all

<0

\( \left\{\begin{array}{c}{\lambda}_3>1,\kern0.5em -1<{\lambda}_4<0\\ {}\frac{{\left(1-{\lambda}_4\right)}^{1-{\lambda}_4}{\left({\lambda}_3-1\right)}^{\lambda_3-1}}{{\left({\lambda}_3-{\lambda}_4\right)}^{\lambda_3-{\lambda}_4}}<\frac{-{\lambda}_4}{\lambda_3}\end{array}\right. \)

λ 1 − (1/λ 2)

3

all

>0

>0

>0

λ 1 − (1/λ 2)

λ 1 + (1/λ 2)

=0

>0

λ 1

λ 1 + (1/λ 2)

>0

=0

λ 1 − (1/λ 2)

λ 1

4

all

<0

<0

<0

−∞

=0

<0

λ 1

<0

=0

−∞

λ 1