cpd: The Complex Triparametric Pearson (CTP) Distribution

cpdR 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) \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.

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:

References

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RCS2003cpd

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RCSO2004cpd

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ROC2018cpd

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COR2021cpd


ujaen-statistics/cpd documentation built on Oct. 2, 2023, 10:43 a.m.