Poisson-Generalized-Exponential Distribution

These functions provide density, distribution function, quantile function, and random number generation for the Poisson-Generalized-Exponential (PGE) Distribution

Usage

dpge(
  x,
  mean = 1,
  shape = 1,
  scale = 1,
  ndraws = 1500,
  log = FALSE,
  haltons = NULL
)

ppge(
  q,
  mean = 1,
  shape = 1,
  scale = 1,
  ndraws = 1500,
  lower.tail = TRUE,
  log.p = FALSE,
  haltons = NULL
)

qpge(p, mean = 1, shape = 1, scale = 1, ndraws = 1500)

rpge(n, mean = 1, shape = 1, scale = 1, ndraws = 1500)

Arguments

x

numeric value or a vector of values.

mean

numeric value or vector of mean values for the distribution (the values have to be greater than 0). This is NOT the value of \(\lambda\).

shape

numeric value or vector of shape values for the shape parameter of the generalized exponential distribution (the values have to be greater than 0).

scale

single value or vector of values for the scale parameter of the generalized exponential distribution (the values have to be greater than 0).

ndraws

the number of Halton draws to use for the integration.

log

logical; if TRUE, probabilities p are given as log(p).

haltons

an optional vector of Halton draws to use instead of ndraws.

q

quantile or a vector of quantiles.

lower.tail

logical; if TRUE, probabilities p are \(P[X\leq x]\) otherwise, \(P[X>x]\).

log.p

logical; if TRUE, probabilities p are given as log(p).

p

probability or a vector of probabilities.

n

the number of random numbers to generate.

Details

dpge computes the density (PDF) of the PGE Distribution.

ppge computes the CDF of the PGE Distribution.

qpge computes the quantile function of the PGE Distribution.

rpge generates random numbers from the PGE Distribution.

The Generalized Exponential distribution can be written as a function with a shape parameter \(\alpha>0\) and scale parameter \(\gamma>0\). The distribution has strictly positive continuous values. The PDF of the distribution is: $$f(x|\alpha,\gamma) = \frac{\alpha}{\gamma} \left(1-e^{-\frac{x}{\gamma}}\right)^{\alpha-1} e^{-\frac{x}{\gamma}}$$

Thus, the compound Probability Mass Function(PMF) for the PGE distribution is: $$f(y|\lambda,\alpha,\beta) = \int_0^\infty \frac{\lambda^y x^y e^{-\lambda x}}{y!} \frac{\alpha}{\gamma} \left(1-e^{-\frac{x}{\gamma}}\right)^{\alpha-1}e^{-\frac{x}{\gamma}} dx$$

The expected value of the distribution is: $$E[y]=\mu=\lambda \left(\frac{\psi(\alpha+1)-\psi(1)}{\gamma}\right)$$

Where \(\psi(\cdot)\) is the digamma function.

The variance is: $$\sigma^2 = \lambda \left(\frac{\psi(\alpha+1)-\psi(1)}{\gamma}\right) + \left(\frac{-\psi'(\alpha+1)+\psi'(1)}{\gamma^2}\right)\lambda^2$$

Where \(\psi'(\cdot)\) is the trigamma function.

To ensure that \(\mu=e^{X\beta}\), \(\lambda\) is replaced with: $$\lambda=\frac{\gamma e^{X\beta}}{\psi(\alpha+1)-\psi(1)}$$

This results in: $$ f(y|\mu,\alpha,\beta) = \int_0^\infty \frac{ \left( \frac{\gamma e^{X\beta}}{\psi(\alpha+1)-\psi(1)} \right)^y x^y e^{ -\left( \frac{\gamma e^{X\beta}}{\psi(\alpha+1)-\psi(1)} \right) x } }{ y! } \frac{\alpha}{\gamma} \left( 1-e^{-\frac{x}{\gamma}} \right)^{\alpha-1} e^{-\frac{x}{\gamma}} dx $$

Halton draws are used to perform simulation over the lognormal distribution to solve the integral.

dpge gives the density, ppge gives the distribution function, qpge gives the quantile function, and rpge generates random deviates.

The length of the result is determined by n for rpge, and is the maximum of the lengths of the numerical arguments for the other functions.

References

Gupta, R. D., & Kundu, D. (2007). Generalized exponential distribution: Existing results and some recent developments. Journal of Statistical planning and inference, 137(11), 3537-3547.

Examples

dpge(0, mean=0.75, shape=2, scale=1, ndraws=2000)
#> [1] 0.5338754
ppge(c(0,1,2,3,4,5,6), mean=0.75, shape=2, scale=1, ndraws=500)
#> [1] 0.5351826 0.8199568 0.9357904 0.9782831 0.9929808 0.9978338 0.9993641
qpge(c(0.1,0.3,0.5,0.9,0.95), mean=0.75, shape=2, scale=1, ndraws=500)
#> [1] 0 0 0 2 3
rpge(30, mean=0.75,  shape=2, scale=1, ndraws=500)
#>  [1] 0 1 0 1 0 0 3 0 2 0 1 0 0 0 0 0 2 1 1 0 3 1 0 0 0 0 0 0 0 0