ctp: The Complex Triparametric Pearson (CTP) Distribution

In ujaen-statistics/cpd: Complex Pearson Distributions

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)

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.

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.

References

\insertRef

RCS2003cpd

\insertRef

RCSO2004cpd

\insertRef

ROC2018cpd

\insertRef

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)

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