Table 2 The support of the random variable T corresponding to the cases (i)-(vi)
From: Families of distributions arising from the quantile of generalized lambda distribution
Case | λ 1 | λ 2 | λ 3 | λ 4 | Q Y (u) | F Y (x) | Support of T |
|---|---|---|---|---|---|---|---|
(i) | free | >0 | >0 | >0 | \( {\lambda}_1+\frac{u^{\lambda_3}-{\left(1-u\right)}^{\lambda_4}}{\lambda_2} \) | Computed numerically. No closed form. | \( \left[{\lambda}_1-{\lambda}_2^{-1},{\lambda}_1+{\lambda}_2^{-1}\right] \) |
(ii) | 1/2 | 2 | >0 | >0 | \( \frac{1+{u}^{\lambda_3}-{\left(1-u\right)}^{\lambda_4}}{2} \) | Computed numerically. No closed form. | [0, 1] |
(iii) | free | <0 | <0 | <0 | \( {\lambda}_1+\frac{u^{\lambda_3}-{\left(1-u\right)}^{\lambda_4}}{\lambda_2} \) | Computed numerically. No closed form. | (−∞, ∞) |
(iv) | free | <0 | =0 | <0 | \( {\lambda}_1+\frac{1-{\left(1-u\right)}^{\lambda_4}}{\lambda_2} \) | \( 1-{\left(1-{\lambda}_2\left(x-{\lambda}_1\right)\right)}^{1/{\lambda}_4} \) | [λ 1, ∞) |
(v) | =0 | <0 | =0 | <0 | \( \left(1-{\left(1-u\right)}^{\lambda_4}\right)/{\lambda}_2 \) | \( 1-{\left(1-{\lambda}_2x\right)}^{1/{\lambda}_4} \) | [0, ∞) |
(vi) | =0 | <0 | <0 | =0 | \( \left({u}^{\lambda_3}-1\right)/{\lambda}_2 \) | \( {\left(1+{\lambda}_2x\right)}^{1/{\lambda}_3} \) | (−∞, 0] |