ebw {cpd}

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)

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.

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.

References

Jose Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ (2003). “A new class of discrete distributions with complex parameters.” Stat. Pap., 44, 67–88. doi:10.1007/s00362-002-0134-7.

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2004). “A triparametric discrete distribution with complex parameters.” Stat. Pap., 45, 81-95. doi:10.1007/BF02778271.

Olmo-Jimenez MJ, Rodriguez-Avi J, Cueva-Lopez V (2018). “A review of the CTP distribution: a comparison with other over- and underdispersed count data models.” Journal of Statistical Computation and Simulation, 88(14), 2684-2706. doi:10.1080/00949655.2018.1482897.

See Also

Functions for maximum-likelihood fitting of the CTP and CBP distributions: fitctp and fitcbp.

Examples

# 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)