Normal mixture with a GPD tail

Spliced bulk-tail family formed by attaching a generalized Pareto tail to a normal mixture bulk. This matches the structure used by the package's normal mixgpd kernels.

Usage

dNormMixGpd(x, w, mean, sd, threshold, tail_scale, tail_shape, log = 0)

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

rNormMixGpd(n, w, mean, sd, threshold, tail_scale, tail_shape)

qNormMixGpd(
  p,
  w,
  mean,
  sd,
  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, sd

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

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 scalar tolerance passed to stats::uniroot.

maxiter

Integer maximum number of iterations for stats::uniroot.

Value

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

Details

If \(F_{mix}\) denotes the normal-mixture CDF, the spliced CDF is $$ F(x) = \left\{ \begin{array}{ll} F_{mix}(x), & x < u, \\ F_{mix}(u) + \{1 - F_{mix}(u)\} G(x), & x \ge u, \end{array} \right. $$ where threshold = u and \(G\) is the GPD CDF.

The construction keeps the normal mixture unchanged below the threshold \(u\) and replaces the upper tail by a generalized Pareto exceedance model. Writing \(F_{mix}(u)=p_u\), the spliced density is $$ f(x) = \left\{ \begin{array}{ll} f_{mix}(x), & x < u, \\ \{1-p_u\} g_{GPD}(x \mid u,\sigma_u,\xi), & x \ge u. \end{array} \right. $$ This formulation preserves total probability because the GPD is attached only to the residual survival mass above the bulk threshold.

The quantile is piecewise. If \(p \le p_u\), qNormMixGpd() inverts the bulk mixture CDF; otherwise it rescales the tail probability to \((p-p_u)/(1-p_u)\) and applies the closed-form GPD quantile. That same piecewise logic is what the fitted-model prediction code uses draw by draw.

Functions

• dNormMixGpd(): Normal mixture + GPD tail density

• pNormMixGpd(): Normal mixture + GPD tail distribution function

• rNormMixGpd(): Normal mixture + GPD tail random generation

• qNormMixGpd(): Normal mixture + GPD tail quantile function

See also

normal_mix(), normal_gpd(), gpd(), normal_lowercase(), dpmgpd().

Other normal kernel families: normal_gpd, normal_mix

Examples

w <- c(0.60, 0.25, 0.15)
mean <- c(-1, 0.5, 2.0)
sd <- c(1.0, 0.7, 1.3)
threshold <- 2
tail_scale <- 1.0
tail_shape <- 0.2

dNormMixGpd(3.0, w, mean, sd, threshold, tail_scale, tail_shape, log = FALSE)
#> [1] 0.0267334
pNormMixGpd(3.0, w, mean, sd, threshold, tail_scale, tail_shape,
           lower.tail = TRUE, log.p = FALSE)
#> [1] 0.9679199
qNormMixGpd(0.50, w, mean, sd, threshold, tail_scale, tail_shape)
#> [1] -0.2713211
qNormMixGpd(0.95, w, mean, sd, threshold, tail_scale, tail_shape)
#> [1] 2.490405
replicate(10, rNormMixGpd(1, w, mean, sd, threshold, tail_scale, tail_shape))
#>  [1]  0.70740840 -1.20709960 -0.29636178 -1.17680354  3.26036430 -1.06133771
#>  [7]  1.29417746 -1.04187750 -0.09906829  2.03944294