ctp | R Documentation |
Probability mass function, distribution function, quantile function and random generation for the Complex Triparametric Pearson (CTP) distribution with parameters a
, b
and \gamma
.
dctp(x, a, b, gamma)
pctp(q, a, b, gamma, lower.tail = TRUE)
qctp(p, a, b, gamma, lower.tail = TRUE)
rctp(n, a, b, gamma)
x |
vector of (non-negative integer) quantiles. |
a |
parameter a (real) |
b |
parameter b (real) |
gamma |
parameter |
q |
vector of quantiles. |
lower.tail |
if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of observations. If |
The CTP distribution with parameters a
, b
and \gamma
has pmf
f(x|a,b,\gamma) = C \Gamma(a+ib+x) \Gamma(a-ib+x) / (\Gamma(\gamma+x) x!), x=0,1,2,...
where i
is the imaginary unit, \Gamma(ยท)
the gamma function and
C = \Gamma(\gamma-a-ib) \Gamma(\gamma-a+ib) / (\Gamma(\gamma-2a) \Gamma(a+ib) \Gamma(a-ib))
the normalizing constant.
If a=0
the CTP is a Complex Biparametric Pearson (CBP) distribution, so the pmf of the CBP distribution is obtained.
If b=0
the CTP is an Extended Biparametric Waring (EBW) distribution, so the pmf of the EBW distribution is obtained.
The mean and the variance of the CTP distribution are
E(X)=\mu=(a^2+b^2)/(\gamma-2a-1)
and Var(X)=\mu(\mu+\gamma-1)/(\gamma-2a-2)
so \gamma > 2a + 2
.
It is underdispersed if a < - (\mu + 1) / 2
, equidispersed if a = - (\mu + 1) / 2
or overdispersed
if a > - (\mu + 1) / 2
. In particular, if a >= 0
the CTP is always overdispersed.
dctp
gives the pmf, pctp
gives the distribution function, qctp
gives the quantile function and rctp
generates random values.
If a = 0
the probability mass function, distribution function, quantile function and random generation function for the CBP distribution arise.
If b = 0
the probability mass function, distribution function, quantile function and random generation function for the EBW distribution arise.
RCS2003cpd
\insertRefRCSO2004cpd
\insertRefROC2018cpd
\insertRefCOR2021cpd
Functions for maximum-likelihood fitting of the CTP, CBP and EBW distributions: fitctp
, fitcbp
and fitebw
.
# Examples for the function dctp
dctp(3,1,2,5)
dctp(c(3,4),1,2,5)
# Examples for the function pctp
pctp(3,1,2,3)
pctp(c(3,4),1,2,3)
# Examples for the function qctp
qctp(0.5,1,2,3)
qctp(c(.8,.9),1,2,3)
# Examples for the function rctp
rctp(10,1,1,3)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.