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 ,
and
.
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 |
q |
vector of quantiles. |
lower.tail |
if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of observations. If |
Details
The CTP distribution with parameters ,
and
has pmf
where is the imaginary unit,
the gamma function and
the normalizing constant.
If the CTP is a Complex Biparametric Pearson (CBP) distribution, so the pmf of the CBP distribution is obtained.
If
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
and
so
.
It is underdispersed if , equidispersed if
or overdispersed
if
. In particular, if
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 the probability mass function, distribution function, quantile function and random generation function for the CBP distribution arise.
If
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)