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Gamma mixture with a GPD tail

gamma_mixgpd.Rd

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

Usage

dGammaMixGpd(x, w, shape, scale, threshold, tail_scale, tail_shape, log = 0)

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

rGammaMixGpd(n, w, shape, scale, threshold, tail_scale, tail_shape)

qGammaMixGpd(
  p,
  w,
  shape,
  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 \(K\).

shape, scale

Numeric vectors of length \(K\) giving Gamma shape and scale parameters.

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 number of iterations for stats::uniroot.

Value

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

Details

The gamma mixture governs the body of the distribution up to the threshold \(u\). Beyond \(u\), only the remaining survival mass is modeled by the GPD, giving $$ f(x) = \left\{ \begin{array}{ll} f_{mix}(x), & x < u, \\ \{1-F_{mix}(u)\} g_{GPD}(x \mid u,\sigma_u,\xi), & x \ge u. \end{array} \right. $$ This is the positive-support analogue of the normal and lognormal splice families. Bulk quantiles are still found by numerical inversion, while tail quantiles use the explicit GPD inverse.

Functions

  • dGammaMixGpd(): Gamma mixture + GPD tail density

  • pGammaMixGpd(): Gamma mixture + GPD tail distribution function

  • rGammaMixGpd(): Gamma mixture + GPD tail random generation

  • qGammaMixGpd(): Gamma mixture + GPD tail quantile function

See also

gamma_mix(), gamma_gpd(), gpd(), gamma_lowercase(), dpmgpd().

Other gamma kernel families: gamma_gpd, gamma_mix

Examples

w <- c(0.55, 0.30, 0.15)
scale <- c(1.0, 2.5, 5.0)
shape <- c(2, 4, 6)
threshold <- 3
tail_scale <- 0.9
tail_shape <- 0.2

dGammaMixGpd(4.0, w = w, scale = scale, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape, log = 0)
#> [1] 0.1831223
pGammaMixGpd(4.0, w = w, scale = scale, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape, lower.tail = 1, log.p = 0)
#> [1] 0.7985655
qGammaMixGpd(0.50, w = w, scale = scale, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape)
#> [1] 3.085593
qGammaMixGpd(0.95, w = w, scale = scale, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape)
#> [1] 5.767674
replicate(10, rGammaMixGpd(1, w = w, scale = scale, shape = shape,
                          threshold = threshold,
                          tail_scale = tail_scale,
                          tail_shape = tail_shape))
#>  [1]  4.803602  2.507495  3.210134 17.763791  3.919686  3.033034  2.196457
#>  [8]  3.964541  3.209980  4.895219