cpd (version 0.3.3)

cbp: The Complex Biparametric Pearson (CBP) Distribution

Description

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

Usage

dcbp(x, b, gamma)
pcbp(q, b, gamma, lower.tail = TRUE)
qcbp(p, b, gamma, lower.tail = TRUE)
rcbp(n, b, gamma)

Value

dcbp gives the pmf, pcbp gives the distribution function, qcbp gives the quantile function and rcbp generates random values.

Arguments

x

vector of (non-negative integer) quantiles.

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 CBP distribution with parameters \(b\) and \(\gamma\) has pmf $$f(x|b,\gamma) = C \Gamma(ib+x) \Gamma(-ib+x) / (\Gamma(\gamma+x) x!), x=0,1,2,...$$ where \(i\) is the imaginary unit, \(\Gamma(·)\) the gamma function and $$C = \Gamma(\gamma-ib) \Gamma(\gamma+ib) / (\Gamma(\gamma) \Gamma(ib) \Gamma(-ib))$$ the normalizing constant.

The CBP is a particular case of the CTP when \(a=0\).

The mean and the variance of the CBP distribution are \(E(X)=\mu=b^2/(\gamma-1)\) and \(Var(X)=\mu(\mu+\gamma-1)/(\gamma-2)\) so \(\gamma > 2\).

It is always overdispersed.

References

RCS2003cpd

See Also

Probability mass function, distribution function, quantile function and random generation for the CTP distribution: dctp, pctp, qctp and rctp. Functions for maximum-likelihood fitting of the CBP distribution: fitcbp.

Examples

# Examples for the function dcbp
dcbp(3,2,5)
dcbp(c(3,4),2,5)

# Examples for the function pcbp
pcbp(3,2,3)
pcbp(c(3,4),2,3)

# Examples for the function qcbp
qcbp(0.5,2,3)
qcbp(c(.8,.9),2,3)

# Examples for the function rcbp
rcbp(10,1,3)