cpd {cpd} | R Documentation |
The Complex Triparametric Pearson (CTP) Distribution
Description
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) doi:10.1007/s00362-002-0134-7, Rodriguez-Avi et al (2004) doi:10.1007/BF02778271 and Olmo-Jimenez et al (2018) doi:10.1080/00949655.2018.1482897. The package also contains maximum-likelihood fitting functions for these models.
Details
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.
Author(s)
Maintainer: Silverio Vilchez-Lopez svilchez@ujaen.es
Authors:
Maria Jose Olmo-Jimenez mjolmo@ujaen.es
Jose Rodriguez-Avi jravi@ujaen.es
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.
Cueva-Lopez V, Olmo-Jimenez MJ, Rodriguez-Avi J (2021). “An Over and Underdispersed Biparametric Extension of the Waring Distribution.” Mathematics, 9(170), 1-15. doi:10.3390/math9020170.