cpd (version 0.3.3)

ctp: The Complex Triparametric Pearson (CTP) Distribution

Description

Probability mass function, distribution function, quantile function and random generation for the Complex Triparametric Pearson (CTP) distribution with parameters \(a\), \(b\) and \(\gamma\).

Usage

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)

Value

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.

Arguments

x

vector of (non-negative integer) quantiles.

a

parameter a (real)

b

parameter b (real)

gamma

parameter \(\gamma\) (positive)

q

vector of quantiles.

lower.tail

if TRUE (default), probabilities are \(P(X<=x)\), otherwise, \(P(X>x)\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

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, \(\Gamma(·)\) 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.

References

RCS2003cpd

RCSO2004cpd

ROC2018cpd

COR2021cpd

See Also

Functions for maximum-likelihood fitting of the CTP, CBP and EBW distributions: fitctp, fitcbp and fitebw.

Examples

# 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)