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 |