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Laplace mixture with a GPD tail

laplace_MixGpd.Rd

Spliced bulk-tail family formed by attaching a generalized Pareto tail to a Laplace mixture bulk.

Usage

dLaplaceMixGpd(
  x,
  w,
  location,
  scale,
  threshold,
  tail_scale,
  tail_shape,
  log = 0
)

pLaplaceMixGpd(
  q,
  w,
  location,
  scale,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = 1,
  log.p = 0
)

rLaplaceMixGpd(n, w, location, scale, threshold, tail_scale, tail_shape)

qLaplaceMixGpd(
  p,
  w,
  location,
  scale,
  threshold,
  tail_scale,
  tail_shape,
  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 \(K\).

location

Numeric vector of length \(K\) giving component locations.

scale

Numeric vector of length \(K\) giving component scales.

threshold

Numeric scalar threshold at which the GPD tail is attached.

tail_scale

Numeric scalar GPD scale parameter; must be positive.

tail_shape

Numeric scalar GPD shape parameter.

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 iterations for stats::uniroot.

Value

Spliced density/CDF/RNG functions return numeric scalars. qLaplaceMixGpd() returns a numeric vector with the same length as p.

Details

This family keeps the Laplace mixture body below the threshold and replaces the upper tail with a generalized Pareto exceedance model scaled by the residual survival mass at the threshold. The tail density is therefore $$ f(x) = \{1-F_{mix}(u)\} g_{GPD}(x \mid u,\sigma_u,\xi), \qquad x \ge u. $$ Bulk quantiles are found numerically from the mixture CDF, and tail quantiles are computed by rescaling the upper-tail probability and applying the GPD inverse.

Functions

  • dLaplaceMixGpd(): Laplace mixture + GPD tail density

  • pLaplaceMixGpd(): Laplace mixture + GPD tail distribution function

  • rLaplaceMixGpd(): Laplace mixture + GPD tail random generation

  • qLaplaceMixGpd(): Laplace mixture + GPD tail quantile function

See also

laplace_mix(), laplace_gpd(), gpd(), laplace_lowercase(), dpmgpd().

Other laplace kernel families: laplace_gpd, laplace_mix

Examples

w <- c(0.50, 0.30, 0.20)
location <- c(-1, 0.5, 2.0)
scale <- c(1.0, 0.7, 1.4)
threshold <- 1
tail_scale <- 1.0
tail_shape <- 0.2

dLaplaceMixGpd(2.0, w = w, location = location, scale = scale,
              threshold = threshold, tail_scale = tail_scale,
              tail_shape = tail_shape, log = FALSE)
#> [1] 0.0865078
pLaplaceMixGpd(2.0, w = w, location = location, scale = scale,
              threshold = threshold, tail_scale = tail_scale,
              tail_shape = tail_shape, lower.tail = TRUE, log.p = FALSE)
#> [1] 0.8961906
qLaplaceMixGpd(0.50, w = w, location = location, scale = scale,
              threshold = threshold, tail_scale = tail_scale,
              tail_shape = tail_shape)
#> [1] -0.02546085
qLaplaceMixGpd(0.95, w = w, location = location, scale = scale,
              threshold = threshold, tail_scale = tail_scale,
              tail_shape = tail_shape)
#> [1] 2.943917
replicate(10, rLaplaceMixGpd(1, w = w, location = location, scale = scale,
                            threshold = threshold,
                            tail_scale = tail_scale,
                            tail_shape = tail_shape))
#>  [1]  0.5347682  2.2306633  1.4405908  2.5020907  6.5262022  6.1675301
#>  [7] -0.5760894 -0.6940654  0.6934115 -1.1575630