cpd | R Documentation |
Probability mass function, distribution function, quantile function and random generation for the Complex Triparametric Pearson (CTP) and Complex Biparametric Pearson (CBP) distributions developed by Rodriguez-Avi et al (2003) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00362-002-0134-7")}, Rodriguez-Avi et al (2004) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02778271")} and Olmo-Jimenez et al (2018) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949655.2018.1482897")}. The package also contains maximum-likelihood fitting functions for these models.
The Complex Triparametric Pearson (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(·)
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.
In this case, a
is call \alpha
.
The mean and the variance of the CTP distribution are
E(X)=\mu=(a^2+b^2)/(\gamma-2a-1)
and Var(X)=E(X)·(E(X)+\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.
Maintainer: Silverio Vilchez-Lopez svilchez@ujaen.es
Authors:
Maria Jose Olmo-Jimenez mjolmo@ujaen.es
Jose Rodriguez-Avi jravi@ujaen.es
RCS2003cpd
\insertRefRCSO2004cpd
\insertRefROC2018cpd
\insertRefCOR2021cpd
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