CCd (version 1.1)

dcc: Density, distribution function and quantile function of the CC distribution

Description

Density, distribution function and quantile function of the CC distribution.

Usage

dcc(y, mu = 0, lambda, logged = FALSE)
pcc(y, mu = 0, lambda)
qcc(p, mu, lambda)

Value

dcc returns a vector with the (logged) density values, the (logged) probabilities for each value of y., pcc returns a vector with the cumulative probabilities, while qcc returns a vector with integer numbers.

Arguments

y

A vector with integer values.

p

A vector with probabilities.

mu

The value of the location parameter \(\mu\).

lambda

The value of the scale parameter \(\lambda\).

logged

Should the logarithm of the density be returned (TRUE) or not (FALSE)?

Author

Michail Tsagris.

R implementation and documentation: Michail Tsagris .

Details

The density of the CC distribution is computed. The probability mass function of the CC distribution (Papadatos, 2022) is given by \(P(X=k)=\dfrac{\tanh{(\lambda \pi)}}{\pi}\dfrac{\lambda}{\lambda^2+\kappa^2}.\)

The cumulative distribution function of the CC distribution is computed. We explore the property of the CC distribution that \(P(X=-\kappa)=P(X=\kappa)\), where \(\kappa>0\), to compute the cumulative distribution.

As for the quantile function we use the optimize function to find the integer whose cumulative probability matches the given probability. So, basically, the qcc() works with left tailed probabilities.

References

Papadatos N. (2022). The characteristic function of the discrete Cauchy distribution In Memory of T. Cacoullos. Journal of Statistical Theory and Practice, 16(3): 47.

See Also

dcc, cc.mle

Examples

x <- round( rcauchy(100, 3, 10) )
mod <- cc.mle(x)
y <- dcc(x, mod$param[1], mod$param[3])

pcc(x[1:5], mod$param[1], mod$param[3])