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Inverse Gaussian with a GPD tail

InvGauss_gpd.Rd

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

Usage

dInvGaussGpd(x, mean, shape, threshold, tail_scale, tail_shape, log = 0)

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

rInvGaussGpd(n, mean, shape, threshold, tail_scale, tail_shape)

qInvGaussGpd(
  p,
  mean,
  shape,
  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.

mean

Numeric scalar mean parameter \(\mu>0\).

shape

Numeric scalar shape parameter \(\lambda>0\).

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. For portability inside NIMBLE, the RNG implementation supports n = 1.

p

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

tol

Numeric tolerance for numerical inversion in qInvGaussGpd.

maxiter

Maximum iterations for numerical inversion in qInvGaussGpd.

Value

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

Details

This is the one-component version of InvGauss_mixgpd(). The inverse Gaussian governs the bulk region and the generalized Pareto governs exceedances over the threshold. The splice is continuous at \(u\) because the GPD is scaled by the inverse-Gaussian survival probability at the threshold.

The ordinary mean of the spliced law exists only when the GPD tail has \(\xi < 1\). When that condition fails, the package uses restricted means or quantile-based summaries instead of an ordinary mean.

Functions

  • dInvGaussGpd(): Inverse Gaussian + GPD tail density

  • pInvGaussGpd(): Inverse Gaussian + GPD tail distribution function

  • rInvGaussGpd(): Inverse Gaussian + GPD tail random generation

  • qInvGaussGpd(): Inverse Gaussian + GPD tail quantile function

See also

InvGauss_mix(), InvGauss_mixgpd(), gpd(), invgauss_lowercase().

Other inverse-gaussian kernel families: InvGauss_mix, InvGauss_mixgpd

Examples

mean <- 2.5
shape <- 6
threshold <- 3
tail_scale <- 0.9
tail_shape <- 0.2

dInvGaussGpd(4.0, mean = mean, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape, log = 0)
#> [1] 0.09181898
pInvGaussGpd(4.0, mean = mean, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape, lower.tail = 1, log.p = 0)
#> [1] 0.8989991
qInvGaussGpd(0.50, mean = mean, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape)
#> [1] 2.077493
qInvGaussGpd(0.95, mean = mean, shape = shape,
            threshold = threshold, tail_scale = tail_scale,
            tail_shape = tail_shape)
#> [1] 4.830437
replicate(10, rInvGaussGpd(1, mean = mean, shape = shape,
                          threshold = threshold,
                          tail_scale = tail_scale,
                          tail_shape = tail_shape))
#>  [1]  0.9172514  2.6620439  4.6838503  1.1727598  1.9694292  3.7568397
#>  [7]  3.7019852  5.1352274 19.0076253  1.1878068