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Gamma with a GPD tail

gamma_gpd.Rd

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

Usage

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

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

rGammaGpd(n, shape, scale, threshold, tail_scale, tail_shape)

qGammaGpd(
  p,
  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.

shape

Numeric scalar Gamma shape parameter.

scale

Numeric scalar scale parameter for the Gamma 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

Integer flag 0/1; if 1, return the log-density.

q

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

lower.tail

Integer flag 0/1; if 1 (default), probabilities are \(P(X \le q)\).

log.p

Integer flag 0/1; if 1, 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 tolerance for numerical inversion in qGammaGpd.

maxiter

Maximum iterations for numerical inversion in qGammaGpd.

Value

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

Details

This topic combines a single gamma bulk with a generalized Pareto exceedance model. If \(F_\Gamma(u)\) is the bulk probability below the threshold, then the splice replaces the upper tail by \(\{1-F_\Gamma(u)\}g_{GPD}(x)\) while leaving the lower region unchanged. The resulting distribution is continuous at the threshold and preserves the gamma body exactly below \(u\).

The ordinary mean is finite only when the GPD shape satisfies \(\xi < 1\). For heavier tails, predictive mean summaries should be replaced by restricted means or quantile summaries.

Functions

  • dGammaGpd(): Gamma + GPD tail density

  • pGammaGpd(): Gamma + GPD tail distribution function

  • rGammaGpd(): Gamma + GPD tail random generation

  • qGammaGpd(): Gamma + GPD tail quantile function

See also

gamma_mix(), gamma_mixgpd(), gpd(), gamma_lowercase().

Other gamma kernel families: gamma_mix, gamma_mixgpd

Examples

scale <- 2.5
shape <- 4
threshold <- 3
tail_scale <- 0.9
tail_shape <- 0.2

dGammaGpd(4.0, scale = scale, shape = shape,
         threshold = threshold, tail_scale = tail_scale,
         tail_shape = tail_shape, log = 0)
#> [1] 0.3220605
pGammaGpd(4.0, scale = scale, shape = shape,
         threshold = threshold, tail_scale = tail_scale,
         tail_shape = tail_shape, lower.tail = 1, log.p = 0)
#> [1] 0.6457335
qGammaGpd(0.50, scale = scale, shape = shape,
         threshold = threshold, tail_scale = tail_scale,
         tail_shape = tail_shape)
#> [1] 3.63375
qGammaGpd(0.95, scale = scale, shape = shape,
         threshold = threshold, tail_scale = tail_scale,
         tail_shape = tail_shape)
#> [1] 6.636445
replicate(10, rGammaGpd(1, scale = scale, shape = shape,
                       threshold = threshold,
                       tail_scale = tail_scale,
                       tail_shape = tail_shape))
#>  [1]  5.468747  3.096501  4.552074 20.631500  6.607379  3.029453  5.115621
#>  [8]  3.217602  6.531285  4.597788