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Cauchy distribution

cauchy.Rd

Scalar Cauchy utilities implemented for NIMBLE compatibility. These functions support symmetric heavy-tailed kernels on the real line and feed directly into the finite Cauchy mixture family.

Usage

dCauchy(x, location, scale, log = 0)

pCauchy(q, location, scale, lower.tail = 1, log.p = 0)

rCauchy(n, location, scale)

qCauchy(p, location, scale, lower.tail = TRUE, log.p = FALSE)

Arguments

x

Numeric scalar giving the point at which the density is evaluated.

location

Numeric scalar location parameter.

scale

Numeric scalar scale parameter; must be positive.

log

Logical; if TRUE, return the log-density (integer flag 0/1 in NIMBLE).

q

Numeric scalar giving the point at which the distribution function is evaluated.

lower.tail

Logical; if TRUE (default), probabilities are \(P(X \le q)\).

log.p

Logical; if TRUE, probabilities are returned on the log scale.

n

Integer giving the number of draws. For portability inside NIMBLE, the RNG implementation supports n = 1.

p

Numeric scalar probability in \((0,1)\) for the quantile function.

Value

dCauchy() returns a numeric scalar density, pCauchy() returns a numeric scalar CDF, rCauchy() returns one random draw, and qCauchy() returns a numeric quantile.

Details

The density is $$ f(x) = \frac{1}{\pi s \{1 + ((x - \ell)/s)^2\}}, $$ where location = ell and scale = s > 0.

These uppercase NIMBLE-compatible functions are scalar (x/q and n = 1). For vectorized R usage, use base_lowercase().

The Cauchy law is a stable heavy-tailed distribution with undefined mean and variance. That is why the package allows the Cauchy kernel only as a bulk distribution and deliberately does not pair it with GPD tails in the kernel registry. For predictive summaries, ordinary means are not available under Cauchy kernels; medians, quantiles, survival curves, and restricted means remain well defined.

The distribution function is $$ F(x) = \frac{1}{2} + \frac{1}{\pi}\arctan\left(\frac{x-\ell}{s}\right), $$ and the quantile is the corresponding inverse $$ Q(p) = \ell + s \tan\{\pi(p-1/2)\}. $$

Functions

  • dCauchy(): Cauchy density function

  • pCauchy(): Cauchy distribution function

  • rCauchy(): Cauchy random generation

  • qCauchy(): Cauchy quantile function

See also

cauchy_mix(), base_lowercase(), kernel_support_table().

Other base bulk distributions: InvGauss, amoroso

Examples

location <- 0
scale <- 1.5

dCauchy(0.5, location, scale, log = 0)
#> [1] 0.1909859
pCauchy(0.5, location, scale, lower.tail = 1, log.p = 0)
#> [1] 0.6024164
qCauchy(0.50, location, scale)
#> [1] 0
qCauchy(0.95, location, scale)
#> [1] 9.470627
replicate(10, rCauchy(1, location, scale))
#>  [1]  1.136833615  9.204051106 -2.358070842 -1.849573362  0.953006225
#>  [6] -0.005439956  0.715536658  0.826377778 -4.820583192  1.654735475