Table 2 Some cdf’s of the new family based on different T distributions.
Distribution of t | cdf of t | G(x) | Range |
|---|---|---|---|
Beta | \(I_{t}(\alpha , \beta )\) | \(I_{x^{\theta }F(x)}(\alpha , \beta )\) | (0, 1) |
Kumaraswamy | \(1-(1-t^{\alpha })^{\beta }\) | \(1-(1-(x^{\theta }F(x))^{\alpha })^{\beta }\) | (0, 1) |
Exponential | \(1- e^{- \alpha t}\) | \(1- e^{-\alpha x^{\theta }F(x)}\) | \((0,\infty )\) |
Weibull | \(1- e^{- \frac{t^{\lambda }}{\beta }}\) | \(1- e^{- \frac{(x^{\theta }F(x))^{\lambda }}{\beta }}\) | \((0,\infty )\) |
Lomax | \(1-(1+\beta t)^{-\alpha }\) | \(1-(1+\beta x^{\theta }F(x))^{-\alpha }\) | \((0,\infty )\) |
Fréchet | \(e^{-\left( \frac{t}{\delta }\right) ^{-\kappa }}\) | \(e^{-\left( \frac{x^{\theta }F(x)}{\delta }\right) ^{-\kappa }}\) | \((0,\infty )\) |
Burr | \(1-(1+ t^{\alpha })^{-\beta }\) | \(1-(1+ (x^{\theta }F(x))^{\alpha })^{-\beta }\) | \((0,\infty )\) |
Log-logistic | \({1}/\left( {1+({t}/{\alpha })^{-\beta }}\right)\) | \({1}/\left( {1+({x^{\theta }F(x)}/{\alpha })^{-\beta }}\right)\) | \((0,\infty )\) |
Gamma | \({\gamma \left( \alpha , \beta t\right) }/{\Gamma (\alpha )}\) | \({\gamma \left( \alpha , \beta x^{\theta }F(x)\right) }/{\Gamma (\alpha )}\) | \((0,\infty )\) |
Log-normal | \(\dfrac{1}{2}\left[ 1+\phi \left( \dfrac{\ln t-\mu }{\sigma \sqrt{2}}\right) \right]\) | \(\dfrac{1}{2}\left[ 1+\phi \left( \dfrac{\ln (x^{\theta }F(x))-\mu }{\sigma \sqrt{2}}\right) \right]\) | \((0,\infty )\) |
