ebw | R Documentation |
Probability mass function, distribution function, quantile function and random generation for the Extended Biparametric Waring (EBW) distribution with parameters \alpha
and \gamma
(or \rho
).
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)
x |
vector of (non-negative integer) quantiles. |
alpha |
parameter alpha (real) |
gamma |
parameter |
rho |
parameter rho (positive) |
q |
vector of quantiles. |
lower.tail |
if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of observations. If |
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(ยท)
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
.
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.
RCS2003cpd
\insertRefRCSO2004cpd
\insertRefROC2018cpd
Functions for maximum-likelihood fitting of the CTP and CBP distributions: fitctp
and fitcbp
.
# 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)
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