Poisson-Weibull Distribution Functions

These functions provide density, distribution function, quantile function, and random number generation for the Poisson-Weibull Distribution, which is specified either by its shape and scale parameters or by its mean and standard deviation.

Usage

dpoisweibull(
  x,
  lambda = NULL,
  alpha = NULL,
  sigma = NULL,
  mean_value = NULL,
  sd_value = NULL,
  ndraws = 1500,
  log = FALSE
)

ppoisweibull(
  q,
  lambda = NULL,
  alpha = NULL,
  sigma = NULL,
  mean_value = NULL,
  sd_value = NULL,
  ndraws = 1500,
  lower.tail = TRUE,
  log.p = FALSE
)

qpoisweibull(
  p,
  lambda = NULL,
  alpha = NULL,
  sigma = NULL,
  mean_value = NULL,
  sd_value = NULL,
  ndraws = 1500
)

rpoisweibull(
  n,
  lambda = NULL,
  alpha = NULL,
  sigma = NULL,
  mean_value = NULL,
  sd_value = NULL,
  ndraws = 1500
)

Arguments

x

A numeric value or vector of values for which the PDF or CDF is calculated.

lambda

Mean value of the Poisson distribution.

alpha

Shape parameter of the Weibull distribution (optional if mean and sd are provided).

sigma

Scale parameter of the Weibull distribution (optional if mean and sd are provided).

mean_value

Mean of the Weibull distribution (optional if alpha and sigma are provided).

sd_value

Standard deviation of the Weibull distribution (optional if alpha and sigma are provided).

ndraws

the number of Halton draws to use for the integration.

log

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

q

Quantile or a vector of quantiles.

lower.tail

Logical; if TRUE, probabilities are P[X <= x], otherwise P[X > x].

log.p

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

p

A numeric value or vector of probabilities for the quantile function.

n

The number of random samples to generate.

Details

The Poisson-Weibull distribution uses the Weibull distribution as a mixing distribution for a Poisson process. It is useful for modeling overdispersed count data. The density function (probability mass function) for the Poisson-Weibull distribution is given by: $$P(y|\lambda,\alpha,\sigma) = \int_0^\infty \frac{e^{-\lambda x} \lambda^y x^y }{y!} \left(\frac{\alpha}{\sigma}\right) \left(\frac{x}{\sigma}\right)^{\alpha-1} e^{-\left(\frac{x}{\sigma}\right)^\alpha} dx$$ where \(f(x| \alpha, \sigma)\) is the PDF of the Weibull distribution and \(\lambda\) is the mean of the Poisson distribution.

dpoisweibull computes the density of the Poisson-Weibull distribution.

ppoisweibull computes the distribution function of the Poisson-Weibull distribution.

qpoisweibull computes the quantile function of the Poisson-Weibull distribution.

rpoisweibull generates random numbers following the Poisson-Weibull distribution.

The shape and scale parameters directly define the Weibull distribution, whereas the mean and standard deviation are used to compute these parameters indirectly.

dpoisweibull gives the density, ppoisweibull gives the distribution function, qpoisweibull gives the quantile function, and rpoisweibull generates random deviates.

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

Examples

dpoisweibull(4, lambda=1.5, mean_value=1.5, sd_value=0.5, ndraws=10)
#> [1] 0.04381581
ppoisweibull(4, lambda=1.5, mean_value=1.5, sd_value=2, ndraws=10)
#> [1] 0.9693094
qpoisweibull(0.95, lambda=1.5, mean_value=1.5, sd_value=2, ndraws=10)
#> [1] 4
rpoisweibull(10, lambda=1.5, mean_value=1.5, sd_value=2, ndraws=10)
#>  [1] 0 1 0 2 2 3 6 0 0 1