ctp {cpd}R Documentation

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 aa, bb and γ\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 γ\gamma (positive)

q

vector of quantiles.

lower.tail

if TRUE (default), probabilities are P(X<=x)P(X<=x), otherwise, P(X>x)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 aa, bb and γ\gamma has pmf

f(xa,b,γ)=CΓ(a+ib+x)Γ(aib+x)/(Γ(γ+x)x!),x=0,1,2,...f(x|a,b,\gamma) = C \Gamma(a+ib+x) \Gamma(a-ib+x) / (\Gamma(\gamma+x) x!), x=0,1,2,...

where ii is the imaginary unit, Γ()\Gamma(·) the gamma function and

C=Γ(γaib)Γ(γa+ib)/(Γ(γ2a)Γ(a+ib)Γ(aib))C = \Gamma(\gamma-a-ib) \Gamma(\gamma-a+ib) / (\Gamma(\gamma-2a) \Gamma(a+ib) \Gamma(a-ib))

the normalizing constant.

If a=0a=0 the CTP is a Complex Biparametric Pearson (CBP) distribution, so the pmf of the CBP distribution is obtained. If b=0b=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)=μ=(a2+b2)/(γ2a1)E(X)=\mu=(a^2+b^2)/(\gamma-2a-1) and Var(X)=μ(μ+γ1)/(γ2a2)Var(X)=\mu(\mu+\gamma-1)/(\gamma-2a-2) so γ>2a+2\gamma > 2a + 2.

It is underdispersed if a<(μ+1)/2a < - (\mu + 1) / 2, equidispersed if a=(μ+1)/2a = - (\mu + 1) / 2 or overdispersed if a>(μ+1)/2a > - (\mu + 1) / 2. In particular, if a>=0a >= 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=0a = 0 the probability mass function, distribution function, quantile function and random generation function for the CBP distribution arise. If b=0b = 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)


[Package cpd version 0.3.2 Index]