Normal with a GPD tail

Spliced family obtained by attaching a generalized Pareto tail above threshold to a single normal bulk component.

Usage

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

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

rNormGpd(n, mean, sd, threshold, tail_scale, tail_shape)

qNormGpd(
  p,
  mean,
  sd,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

x

Numeric scalar giving the point at which the density is evaluated.

mean

Numeric scalar mean parameter for the Normal bulk.

sd

Numeric scalar standard deviation for the Normal 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.

Value

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

Details

This is the single-component version of normal_mixgpd(). If \(\Phi_u = \Phi((u-\mu)/\sigma)\) denotes the normal bulk probability below the threshold, then the density is $$ f(x) = \left\{ \begin{array}{ll} \phi(x \mid \mu, \sigma^2), & x < u, \\ (1-\Phi_u) g_{GPD}(x \mid u,\sigma_u,\xi), & x \ge u. \end{array} \right. $$ The distribution is continuous at \(u\) by construction, although the derivative generally changes there because the tail is modeled by a different family.

The ordinary mean exists only when the GPD tail satisfies \(\xi < 1\). When that condition fails, downstream mean prediction is intentionally blocked and the package directs the user to restricted means or quantile-based summaries instead.

Functions

• dNormGpd(): Normal + GPD tail density

• pNormGpd(): Normal + GPD tail distribution function

• rNormGpd(): Normal + GPD tail random generation

• qNormGpd(): Normal + GPD tail quantile function

See also

normal_mix(), normal_mixgpd(), gpd(), normal_lowercase().

Other normal kernel families: normal_mix, normal_mixgpd

Examples

mean <- 0.5
sd <- 1.0
threshold <- 2
tail_scale <- 1.0
tail_shape <- 0.2

dNormGpd(3.0, mean, sd, threshold, tail_scale, tail_shape, log = FALSE)
#> [1] 0.0223736
pNormGpd(3.0, mean, sd, threshold, tail_scale, tail_shape,
        lower.tail = TRUE, log.p = FALSE)
#> [1] 0.9731517
qNormGpd(0.50, mean, sd, threshold, tail_scale, tail_shape)
#> [1] 0.5
qNormGpd(0.95, mean, sd, threshold, tail_scale, tail_shape)
#> [1] 2.29835
replicate(10, rNormGpd(1, mean, sd, threshold, tail_scale, tail_shape))
#>  [1]  0.81316615  0.53476606 -0.51398676  0.50758678 -0.06515087  0.83740151
#>  [7]  0.82488512  2.34031674  0.85547666  2.53418989