On generating T-X family of distributions using quantile functions

Aljarrah, Mohammad A; Lee, Carl; Famoye, Felix

doi:10.1186/2195-5832-1-2

Journal of Statistical Distributions and Applications

Table 1 T-X { Y } families based on different quantile functions

From: On generating T-X family of distributions using quantile functions

Random variable Y and support of r( t)	The quantile function Q_Y( λ)	Family of probability density function g( x) defined in (2.4)
Exponential (0, ∞)	- b log(1 - λ), b > 0	$\frac{bf (x)}{1 - F (x)} r \{- b log (1 - F (x))\}$
Weibull (0, ∞)	γ{-log(1 - λ)}^1/c, γ, c > 0	$\frac{γ f (x) r \{γ {\{- log (1 - F (x))\}}^{1 / c}\}}{c (1 - F (x)) {\{- log (1 - F (x))\}}^{(c - 1) / c}}$
Rayleigh (0, ∞)	{-2b² log(1 - λ)}^1/2, b > 0	$\frac{bf (x) r \{{\{- 2 b^{2} log (1 - F (x))\}}^{1 / 2}\}}{(1 - F (x)) {\{- 2 log (1 - F (x))\}}^{1 / 2}}$
Dagum (0, ∞)	$β {\{\frac{λ^{1 / p}}{1 - λ^{1 / p}}\}}^{1 / α}, α, β, p > 0$	$\frac{β f (x) r \{β F^{1 / α p} (x) / {(1 - F^{1 / p} (x))}^{1 / α}\}}{α p F^{1 - 1 / α p} (x) {(1 - F^{1 / p} (x))}^{1 + 1 / α}}$
Lomax (0, ∞)	$\frac{1}{α} \{\frac{1 - {(1 - λ)}^{1 / k}}{{(1 - λ)}^{1 / k}}\}, α, k > 0$	$\frac{f (x)}{k α {(1 - F (x))}^{1 / k + 1}} r \{\frac{1 - {(1 - F (x))}^{1 / k}}{α {(1 - F (x))}^{1 / k}}\}$
Log-logistic (0, ∞)	$α {(\frac{λ}{1 - λ})}^{1 / β}, α, β > 0$	$\frac{α f (x) r \{α {(F (x) / \{1 - F (x)\})}^{1 / β}\}}{β F^{(β - 1) / β} (x) {(1 - F (x))}^{(β + 1) / β}}$
Exponentiated Exponential (0, ∞)	$- \frac{1}{θ} log (1 - λ^{1 / α}), θ, α > 0$	$\frac{f (x) r \{- (1 / θ) log (1 - F^{1 / α} (x))\}}{α θ F^{(α - 1) / α} (x) (1 - F^{1 / α} (x))}$
Cauchy (-∞, ∞)	a + b {tan(π(λ - 0.5))}, b > 0	$\frac{π b f (x) r \{a + b tan \{π (F (x) - 1 / 2)\}\}}{{cos}^{2} \{π (F (x) - 1 / 2)\}}$
Extreme value(Gumbel) (-∞, ∞)	a - b log(-log λ), b > 0	$\frac{bf (x) r \{a - b log (- log F (x))\}}{- F (x) log F (x)}$
Laplace (-∞, ∞)	$\begin{array}{l} \{\begin{array}{c} a + b log (2 λ), λ < 0.5 \\ a - b log (2 (1 - λ)), λ \geq 0.5 \end{array} \\ a, b > 0 \end{array}$	$\{\begin{array}{c} \frac{bf (x) r \{a + b log (2 F (x))\}}{F (x)}, F (x) < .5 \\ \frac{bf (x) r \{a - b log \{2 (1 - F (x))\}\}}{1 - F (x)}, F (x) \geq . 5 \end{array}$
Logistic (-∞, ∞)	$a + b log (\frac{λ}{1 - λ}), b > 0$	$\frac{bf (x) r \{a + b log (F (x) / \{1 - F (x)\})\}}{F (x) (1 - F (x))}$
Generalized logistic (II) (-∞, ∞)	$log \{\frac{1 - {(1 - λ)}^{1 / α}}{{(1 - λ)}^{1 / α}}\}, α > 0$	$\frac{f (x) r \{log (\{1 - {(1 - F (x))}^{1 / α}\} / {(1 - F (x))}^{1 / α})\}}{α (1 - F (x)) \{1 - {(1 - F (x))}^{1 / α}\}}$

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