ebw: The Extended Biparametric Waring (EBW) Distribution

Description

Probability mass function, distribution function, quantile function and random generation for the Extended Biparametric Waring (EBW) distribution with parameters $\alpha$ and $\gamma$ (or $\rho$).

Usage

debw(x, alpha, gamma, rho)
pebw(q, alpha, gamma, rho, lower.tail = TRUE)
qebw(p, alpha, gamma, rho, lower.tail = TRUE)
rebw(n, alpha, gamma, rho, lower.tail = TRUE)

Value

debw gives the pmf, pebw gives the distribution function, qebw gives the quantile function and rebw generates random values.

If $\alpha > 0$ the probability mass function, distribution function, quantile function and random generation function for the UGW$(\alpha,\alpha,\rho)$ distribution arise.

If $\alpha < 0$ the probability mass function, distribution function, quantile function and random generation function for the CTP$(\alpha,0,\gamma)$ distribution arise.

Arguments

x: vector of (non-negative integer) quantiles.
alpha: parameter alpha (real)
gamma: parameter $\gamma$ (positive)
rho: parameter rho (positive)
q: vector of quantiles.
lower.tail: if TRUE (default), probabilities are $P(X<=x)$, otherwise, $P(X>x)$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

The EBW distribution with parameters $\alpha$ and $\gamma$ has pmf $$f(x|a,\alpha,\gamma) = C \Gamma(\alpha+x)^2 / (\Gamma(\gamma+x) x!), x=0,1,2,...$$ where $\Gamma(·)$ is the gamma function and $$C = \Gamma(\gamma-\alpha^2 / (\Gamma(\alpha)^2 \Gamma(\gamma-2a))$$ the normalizing constant.

There is an alternative parametrization in terms of $\alpha$ and $\rho=\gamma-2\alpha>0$ when $\alpha>0$. So, introduce only $\alpha$ and $\gamma$ or $\alpha$ and $\rho$, depending on the parametrization you wish to use.

The mean and the variance of the EBW distribution are $E(X)=\mu=\alpha^2/(\gamma-2\alpha-1)$ and $Var(X)=\mu(\mu+\gamma-1)/(\gamma-2\alpha-2)$ so $\gamma > 2a + 2$.

It is underdispersed if $\alpha < - (\mu + 1) / 2$, equidispersed if $\alpha = - (\mu + 1) / 2$ or overdispersed if $\alpha > - (\mu + 1) / 2$. In particular, if $\alpha >= -0.5$ the EBW is overdispersed, whereas if $\alpha < -1$ the EBW is underdispersed. In the case $-1 < \alpha <= -0.5$, the EBW may be under-, equi- or overdispersed depending on the value of $\gamma$.

References

RCS2003cpd

RCSO2004cpd

ROC2018cpd

Examples

Run this code

# Examples for the function dctp
debw(3,1,rho=5)
debw(c(3,4),2,rho=5)

# Examples for the function pebw
pebw(3,2,rho=5)
pebw(c(3,4),2,rho=5)

# Examples for the function qebw
qebw(0.5,-2.1,gamma=0.1)
qebw(c(.8,.9),-2.1,gamma=0.1)
qebw(0.5,2,rho=5)
qebw(c(.8,.9),2,rho=5)

# Examples for the function rebw
rebw(10,2,rho=5)
rebw(10,-2.1,gamma=5)

Run the code above in your browser using DataLab