In Auburngrads/SMRD: Statistical Methods For Reliability Data

Functional relationships for $\;T \sim GENG(\theta,\beta,\kappa),\;\;T \in [0,\infty),\;\;\; \theta,\beta,\kappa\; > 0$

$$ \begin{aligned} f(t|\theta,\beta,\kappa)&=\Gamma_{I}\left[\left(\frac{t}{\theta}\right)^{\beta},\kappa\right]\\\\ F(t|\theta,\beta,\kappa)&=\frac{\beta}{\Gamma(\kappa)\theta}\left(\frac{t}{\theta}\right)^{\kappa\beta-1}\exp\left[-\left(\frac{t}{\theta}\right)^{\beta}\right]\\\\ t_{p}&=\theta \left[\Gamma_{I}^{-1}(p;\kappa)\right]^{1/\beta}\\\\ E[T]&=\frac{\theta\;\Gamma(1/\beta)+\kappa}{\Gamma(\kappa)}\\\\ Var[T]&=\theta^2\left[\frac{\Gamma(2/\beta+\kappa)}{\Gamma(\kappa)}-\frac{\Gamma^{2}(1/\beta)+\kappa}{\Gamma^{2}(\kappa)}\right] \end{aligned} $$

• $\theta \in \mathbb{R}^{+}$ is a scale parameter

• $\beta \in \mathbb{R}^{+}$ is a shape parameter

• $\kappa \in \mathbb{R}^{+}$ is a shape parameter

• $\kappa \in \mathbb{R}^{+}$ is a shape parameter

• $\theta \in \mathbb{R}^{+}$ is a scale parameter

• $\Gamma(z)$ is the gamma function defined as

$$ \Gamma(z) = \begin{cases} \int_0^{\infty} x^{z-1}e^{-x}dx \hspace{12pt}\text{ if } z \in \mathbb{R}\\ (z - 1)! \hspace{40pt} \mbox{ if } z \in \mathbb{I} \end{cases} $$

• $\Gamma_{I}(a,b)$ is the lower incomplete gamma function defined as

$$ \Gamma_{I}(a,b) = \int_{0}^{b} t^{a-1}e^{-t}dt. $$

• $\Gamma_{I}^{-1}(a,b)$ is the inverse of the lower incomplete gamma function defined as

$$ \Gamma_{I}^{-1}(a,b) = \frac{\Gamma_{I}(b,a)}{\Gamma(b)} = \frac{1}{\Gamma(b)}\int_{0}^{a} t^{b-1}e^{-t}dt. $$

• $\Gamma(z)$ values can be computed in R using the base function gamma(x)

• $\Gamma_{I}(a,b)$ values can be computed using the gamma_inc(a,b) function from the gsl package

• $\Gamma_{I}^{-1}(a,b)$ values can be computed using the gamma_inc_P(a,b) function from the gsl package

Auburngrads/SMRD documentation built on Sept. 14, 2020, 2:21 a.m.