Cauchy: Cauchy Distribution

Description

The Cauchy distribution is an absolute continuous probability distribution characterized by its location parameter $x_0$ and scale parameter $\gamma > 0$.

Usage

Cauchy(location = 0, scale = 1)
# S4 method for Cauchy,numeric
d(distr, x, log = FALSE)
# S4 method for Cauchy,numeric
p(distr, q, lower.tail = TRUE, log.p = FALSE)
# S4 method for Cauchy,numeric
qn(distr, p, lower.tail = TRUE, log.p = FALSE)
# S4 method for Cauchy,numeric
r(distr, n)
# S4 method for Cauchy
mean(x)
# S4 method for Cauchy
median(x)
# S4 method for Cauchy
mode(x)
# S4 method for Cauchy
var(x)
# S4 method for Cauchy
sd(x)
# S4 method for Cauchy
skew(x)
# S4 method for Cauchy
kurt(x)
# S4 method for Cauchy
entro(x)
# S4 method for Cauchy
finf(x)
llcauchy(x, location, scale)
# S4 method for Cauchy,numeric
ll(distr, x)
ecauchy(x, type = "mle", ...)
# S4 method for Cauchy,numeric
mle(
  distr,
  x,
  par0 = "me",
  method = "L-BFGS-B",
  lower = c(-Inf, 1e-05),
  upper = c(Inf, Inf),
  na.rm = FALSE
)
# S4 method for Cauchy,numeric
me(distr, x, na.rm = FALSE)
vcauchy(location, scale, type = "mle")
# S4 method for Cauchy
avar_mle(distr)

Value

Each type of function returns a different type of object:

Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.
Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.
Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Arguments

location, scale: numeric. Location and scale parameters.
distr: an object of class Cauchy.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Cauchy. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p: logical. Should the logarithm of the probability be returned?
q: numeric. Vector of quantiles.
lower.tail: logical. If TRUE (default), probabilities are $P(X \leq x)$, otherwise $P(X > x)$.
p: numeric. Vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
type: character, case ignored. The estimator type (mle or me).
...: extra arguments.
par0, method, lower, upper: arguments passed to optim for the mle optimization.
na.rm: logical. Should the NA values be removed?

Details

The probability density function (PDF) of the Cauchy distribution is given by: $$ f(x; x_0, \gamma) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]}.$$

The MLE of the Cauchy distribution parameters is not available in closed form and has to be approximated numerically. This is done with optim(). The default method used is the L-BFGS-B method with lower bounds c(-Inf, 1e-5) and upper bounds c(Inf, Inf). The par0 argument can either be a numeric (both elements satisfying lower <= par0 <= upper) or a character specifying the closed-form estimator to be used as initialization for the algorithm ("me" - the default value).

Note that the me() estimator for the Cauchy distribution is not a moment estimator; it utilizes the sample median instead of the sample mean.

Examples

Run this code

# -----------------------------------------------------
# Cauchy Distribution Example
# -----------------------------------------------------

# Create the distribution
x0 <- 3 ; scale <- 5
D <- Cauchy(x0, scale)

# ------------------
# dpqr Functions
# ------------------

d(D, c(-5, 3, 10)) # density function
p(D, c(-5, 3, 10)) # distribution function
qn(D, c(0.4, 0.8)) # inverse distribution function
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

median(D) # Median
mode(D) # Mode
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llcauchy(x, x0, scale)

ecauchy(x, type = "mle")
ecauchy(x, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("cauchy", x) # the distr argument can be a character

# ------------------
# Estimator Variance
# ------------------

vcauchy(x0, scale, type = "mle")
avar_mle(D)
v(D, type = "mle")

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