Inverse Gaussian mixture with a GPD tail

Spliced bulk-tail family formed by attaching a generalized Pareto tail to an inverse Gaussian mixture bulk.

Usage

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

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

rInvGaussMixGpd(n, w, mean, shape, threshold, tail_scale, tail_shape)

qInvGaussMixGpd(
  p,
  w,
  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.

w

Numeric vector of mixture weights of length \(K\).

mean, shape

Numeric vectors of length \(K\) giving component means and shapes.

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 scalar tolerance passed to stats::uniroot in quantile inversion.

maxiter

Integer maximum number of iterations for stats::uniroot.

Value

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

Details

This family keeps the inverse-Gaussian mixture body below the threshold \(u\) and attaches a generalized Pareto exceedance law to the residual survival probability above \(u\). If \(F_{mix}(u)=p_u\), then the tail density is \((1-p_u)g_{GPD}(x \mid u,\sigma_u,\xi)\).

Quantile evaluation is piecewise. For probabilities at or below \(p_u\), the function solves the mixture inverse numerically; above \(p_u\), it rescales the upper-tail probability and applies the GPD inverse directly.

Functions

• dInvGaussMixGpd(): Inverse Gaussian mixture + GPD tail density

• pInvGaussMixGpd(): Inverse Gaussian mixture + GPD tail distribution function

• rInvGaussMixGpd(): Inverse Gaussian mixture + GPD tail random generation

• qInvGaussMixGpd(): Inverse Gaussian mixture + GPD tail quantile function

See also

InvGauss_mix(), InvGauss_gpd(), gpd(), invgauss_lowercase(), dpmgpd().

Other inverse-gaussian kernel families: InvGauss_gpd, InvGauss_mix

Examples

w <- c(0.55, 0.30, 0.15)
mean <- c(1.0, 2.5, 5.0)
shape <- c(2, 4, 8)
threshold <- 3
tail_scale <- 0.9
tail_shape <- 0.2

dInvGaussMixGpd(4.0, w = w, mean = mean, shape = shape,
               threshold = threshold, tail_scale = tail_scale,
               tail_shape = tail_shape, log = 0)
#> [1] 0.06286562
pInvGaussMixGpd(4.0, w = w, mean = mean, shape = shape,
               threshold = threshold, tail_scale = tail_scale,
               tail_shape = tail_shape, lower.tail = 1, log.p = 0)
#> [1] 0.9308478
qInvGaussMixGpd(0.50, w = w, mean = mean, shape = shape,
               threshold = threshold, tail_scale = tail_scale,
               tail_shape = tail_shape)
#> [1] 1.251694
qInvGaussMixGpd(0.95, w = w, mean = mean, shape = shape,
               threshold = threshold, tail_scale = tail_scale,
               tail_shape = tail_shape)
#> [1] 4.368537
replicate(10, rInvGaussMixGpd(1, w = w, mean = mean, shape = shape,
                             threshold = threshold,
                             tail_scale = tail_scale,
                             tail_shape = tail_shape))
#>  [1] 0.3554396 0.6082587 2.3634573 0.7692852 5.0918843 1.8738874 1.0267480
#>  [8] 0.8791639 0.4361379 0.5604970