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) tools:::Rd_expr_doi("10.1007/s00362-002-0134-7"), Rodriguez-Avi et al (2004) tools:::Rd_expr_doi("10.1007/BF02778271") and Olmo-Jimenez et al (2018) tools:::Rd_expr_doi("10.1080/00949655.2018.1482897"). The package also contains maximum-likelihood fitting functions for these models.
Maintainer: Silverio Vilchez-Lopez svilchez@ujaen.es
Authors:
Maria Jose Olmo-Jimenez mjolmo@ujaen.es
Jose Rodriguez-Avi jravi@ujaen.es
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.
RCS2003cpd
RCSO2004cpd
ROC2018cpd
COR2021cpd