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Cauchy mixture distribution

cauchy_mix.Rd

Finite mixture of Cauchy components for symmetric heavy-tailed bulk modeling on the real line.

Usage

dCauchyMix(x, w, location, scale, log = 0)

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

rCauchyMix(n, w, location, scale)

qCauchyMix(
  p,
  w,
  location,
  scale,
  lower.tail = TRUE,
  log.p = FALSE,
  tol = 1e-10,
  maxiter = 200
)

Arguments

x

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

w

Numeric vector of mixture weights of length \(K\). The functions normalize w internally when needed.

location, scale

Numeric vectors of length \(K\) giving component locations and scales.

log

Logical; if TRUE, return the log-density.

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. The RNG implementation supports n = 1.

p

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

tol

Numeric scalar tolerance passed to stats::uniroot.

maxiter

Integer maximum number of iterations for stats::uniroot.

Value

Density/CDF/RNG functions return numeric scalars. qCauchyMix() returns a numeric vector with the same length as p.

Details

The mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k f_C(x \mid \ell_k, s_k), $$ with normalized weights \(\tilde{w}_k\). These scalar functions are NIMBLE-compatible; for vectorized R usage, use cauchy_mix_lowercase().

The mixture CDF is the weighted average of component CDFs, $$ F(x) = \sum_{k=1}^K \tilde{w}_k \left\{\frac{1}{2} + \frac{1}{\pi}\arctan\left(\frac{x-\ell_k}{s_k}\right)\right\}. $$ Random generation first selects a component according to the normalized weights and then draws from the chosen Cauchy law by inverse-CDF sampling.

Because each Cauchy component has undefined mean and variance, the mixture also lacks an ordinary mean in general. That is why the package exposes Cauchy kernels for densities, CDFs, quantiles, medians, survival functions, and restricted means, but not for ordinary predictive means.

Functions

  • dCauchyMix(): Cauchy mixture density

  • pCauchyMix(): Cauchy mixture distribution function

  • rCauchyMix(): Cauchy mixture random generation

  • qCauchyMix(): Cauchy mixture quantile function

See also

cauchy(), cauchy_mix_lowercase(), build_nimble_bundle(), kernel_support_table().

Examples

w <- c(0.50, 0.30, 0.20)
location <- c(-2, 0, 3)
scale <- c(1.0, 0.7, 1.5)

dCauchyMix(0.5, w = w, location = location, scale = scale, log = FALSE)
#> [1] 0.1235181
pCauchyMix(0.5, w = w, location = location, scale = scale,
           lower.tail = TRUE, log.p = FALSE)
#> [1] 0.6830742
qCauchyMix(0.50, w = w, location = location, scale = scale)
#> [1] -0.6639507
qCauchyMix(0.95, w = w, location = location, scale = scale)
#> [1] 6.996407
replicate(10, rCauchyMix(1, w = w, location = location, scale = scale))
#>  [1] -2.5795875 -0.6903653  1.7792099  5.3486299  3.3920039 45.6839639
#>  [7] -0.8740658 -1.1633733  1.3870253  3.5761715