ctp {cpd}

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

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.

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2004). “A triparametric discrete distribution with complex parameters.” Stat. Pap., 45, 81-95. doi:10.1007/BF02778271.

Olmo-Jimenez MJ, Rodriguez-Avi J, Cueva-Lopez V (2018). “A review of the CTP distribution: a comparison with other over- and underdispersed count data models.” Journal of Statistical Computation and Simulation, 88(14), 2684-2706. doi:10.1080/00949655.2018.1482897.

Cueva-Lopez V, Olmo-Jimenez MJ, Rodriguez-Avi J (2021). “An Over and Underdispersed Biparametric Extension of the Waring Distribution.” Mathematics, 9(170), 1-15. doi:10.3390/math9020170.

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)