In teachingApps: Apps for Teaching Statistics, R Programming, and Shiny App Development

Functional relationships for $\;X \sim EGENG(\mu,\sigma,\lambda),\;\;T \in [0,\infty),\;\;\; \mu \in (-\infty,\infty),\sigma \in (0,\infty),\lambda \in (-\infty, \infty)\; > 0$

$$ \begin{aligned} f(t|\mu,\sigma,\lambda)&=\begin{cases} \frac{\lambda}{\sigma t} \phi_{lg}\left[\lambda\omega+ \ln(\lambda^{-2}); \lambda^{-2}\right] &\mbox{if } \lambda \ne 0\\ \frac{1}{\sigma t} \phi_{nor}(\omega) &\mbox{if } \lambda = 0 \end{cases}\\ F(t|\mu,\sigma,\lambda)&=\begin{cases}\Phi_{lg}\left[\lambda\omega+ \ln(\lambda^{-2}); \lambda^{-2}\right] &\mbox{if } \lambda > 0\\ \Phi_{nor}(\omega) &\mbox{if } \lambda = 0\\ 1-\Phi_{lg}\left[\lambda\omega+\ln(\lambda^{-2});\lambda^{-2}\right]&\mbox{if }\lambda <0\end{cases}\\ t_{p}&=\exp\left[\mu+\sigma\times \omega(p;\lambda)\right]\\ E[T]&=\exp(\mu+0.5\sigma^2)\\ Var[T]&=\exp(2\mu+\sigma^2)(\exp(\sigma^2)-1) \end{aligned} $$

• $\mu \in \mathbb{R}$ is a scale parameter

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

• $\lambda \in \mathbb{R}$ is a shape parameter

• $\phi_{lg}(z;a)$ is the standardized pdf of the log-gamma distribution

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

• $\Gamma_{I}(a,b)$ 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

teachingApps documentation built on July 1, 2020, 5:58 p.m.