cbp {cpd}

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)

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.

Value

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

References

Jose Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ (2003). “A new class of discrete distributions with complex parameters.” Stat. Pap., 44, 67–88. doi:10.1007/s00362-002-0134-7.

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)