lca: The Log-Cauchy Distribution

In rdmatheus/BCNSM: Multivariate Box-Cox Symmetric Distributions Generated by a Normal Scale Mixture Copula

The Log-Cauchy Distribution

Description

Density, distribution function, quantile function, and random generation for the log-Cauchy distribution with parameters mu and sigma.

Usage

dlca(x, mu, sigma, log = FALSE, ...)

plca(q, mu, sigma, lower.tail = TRUE, ...)

qlca(p, mu, sigma, lower.tail = TRUE, ...)

rlca(n, mu, sigma)

Arguments

x, q	vector of positive quantiles.
mu	vector of strictly positive scale parameters.
sigma	vector of strictly positive relative dispersion parameters.
log	logical; if TRUE, probabilities p are given as log(p).
...	further arguments.
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p	vector of probabilities.
n	number of random values to return.

Details

A random variable X has a log-Cauchy distribution with parameter mu and sigma if log(X) follows a Cauchy distribution with location parameter log(mu) and dispersion parameter sigma. It can be showed that mu is the median of X.

Value

dlca returns the density, plca gives the distribution function, qlca gives the quantile function, and rlca generates random observations.

Invalid arguments will result in return value NaN.

The length of the result is determined by n for rlca, and is the maximum of the lengths of the numerical arguments for the other functions.

Author(s)

Rodrigo M. R. de Medeiros <[e-mail]>

References

Vanegas, L. H., and Paula, G. A. (2016). Log-symmetric distributions: statistical properties and parameter estimation. Brazilian Journal of Probability and Statistics, 30, 196-220.

Examples

mu <- 2
sigma <- 1

# Sample generation
x <- rlca(1000, mu, sigma)

# Density
hist(x, prob = TRUE, main = "The Log-Cauchy Distribution", col = "white")
curve(dlca(x, mu, sigma), add = TRUE, col = 2, lwd = 2)
legend("topright", "Probability density function", col = 2, lwd = 2, lty = 1)

# Distribution function
plot(ecdf(x), main = "The Log-Cauchy Distribution", ylab = "Distribution function")
curve(plca(x, mu, sigma), add = TRUE, col = 2, lwd = 2)
legend("bottomright", c("Emp. distribution function", "Theo. distribution function"),
  col = c(1, 2), lwd = 2, lty = c(1, 1)
)

# Quantile function
plot(seq(0.01, 0.99, 0.001), quantile(x, seq(0.01, 0.99, 0.001)),
  type = "l",
  xlab = "p", ylab = "Quantile function", main = "The Log-Cauchy Distribution"
)
curve(qlca(x, mu, sigma), add = TRUE, col = 2, lwd = 2, from = 0, to = 1)
legend("topleft", c("Emp. quantile function", "Theo. quantile function"),
  col = c(1, 2), lwd = 2, lty = c(1, 1)
)

rdmatheus/BCNSM documentation built on Feb. 8, 2024, 1:28 a.m.