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Laplace with a GPD tail

laplace_gpd.Rd

Spliced family obtained by attaching a generalized Pareto tail above threshold to a single Laplace bulk.

Usage

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

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

rLaplaceGpd(n, location, scale, threshold, tail_scale, tail_shape)

qLaplaceGpd(
  p,
  location,
  scale,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

x

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

location

Numeric scalar location parameter for the Laplace bulk.

scale

Numeric scalar scale parameter for the Laplace bulk.

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.

Value

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

Details

This topic pairs a single Laplace bulk distribution with a generalized Pareto exceedance tail. The splice is continuous at the threshold because the tail density is multiplied by the Laplace survival probability at that threshold.

The ordinary mean exists only when the GPD tail has \(\xi < 1\). If the fitted tail is too heavy for an ordinary mean to exist, the package directs users to restricted means or quantiles rather than returning an unstable mean summary.

Functions

  • dLaplaceGpd(): Laplace + GPD tail density

  • pLaplaceGpd(): Laplace + GPD tail distribution function

  • rLaplaceGpd(): Laplace + GPD tail random generation

  • qLaplaceGpd(): Laplace + GPD tail quantile function

See also

laplace_mix(), laplace_MixGpd(), gpd(), laplace_lowercase().

Other laplace kernel families: laplace_MixGpd, laplace_mix

Examples

location <- 0.5
scale <- 1.0
threshold <- 1
tail_scale <- 1.0
tail_shape <- 0.2

dLaplaceGpd(2.0, location, scale, threshold, tail_scale, tail_shape, log = FALSE)
#> [1] 0.1015629
pLaplaceGpd(2.0, location, scale, threshold, tail_scale, tail_shape,
           lower.tail = TRUE, log.p = FALSE)
#> [1] 0.8781245
qLaplaceGpd(0.50, location, scale, threshold, tail_scale, tail_shape)
#> [1] 0.5
qLaplaceGpd(0.95, location, scale, threshold, tail_scale, tail_shape)
#> [1] 3.170353
replicate(10, rLaplaceGpd(1, location, scale, threshold, tail_scale, tail_shape))
#>  [1]  1.26300782  0.64116940 -0.45552592  0.19260121  1.02705848  2.79357033
#>  [7] -0.24057496 -1.34531772  0.04364005  1.82800703