# ctp: The Complex Triparametric Pearson (CTP) Distribution In cpd: Complex Pearson Distributions

 ctp 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 a, b and γ.

### 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 (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 γ has pmf

f(x|a,b,γ) = C Γ(a+ib+x) Γ(a-ib+x) / (Γ(γ+x) x!), x=0,1,2,...

where i is the imaginary unit, Γ(·) the gamma function and

C = Γ(γ-a-ib) Γ(γ-a+ib) / (Γ(γ-2a) Γ(a+ib) Γ(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)=μ=(a^2+b^2)/(γ-2a-1) and Var(X)=μ(μ+γ-1)/(γ-2a-2) so γ > 2a + 2.

It is underdispersed if a < - (μ + 1) / 2, equidispersed if a = - (μ + 1) / 2 or overdispersed if a > - (μ + 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

\insertRef

RCS2003cpd

\insertRef

RCSO2004cpd

\insertRef

ROC2018cpd

\insertRef

COR2021cpd

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)

```

cpd documentation built on Aug. 9, 2022, 5:08 p.m.