Lognormal with a GPD tail

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

Usage

dLognormalGpd(x, meanlog, sdlog, threshold, tail_scale, tail_shape, log = 0)

pLognormalGpd(
  q,
  meanlog,
  sdlog,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = 1,
  log.p = 0
)

rLognormalGpd(n, meanlog, sdlog, threshold, tail_scale, tail_shape)

qLognormalGpd(
  p,
  meanlog,
  sdlog,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

x

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

meanlog

Numeric scalar log-mean parameter for the Lognormal bulk.

sdlog

Numeric scalar log-standard deviation for the Lognormal 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. qLognormalGpd() returns a numeric vector with the same length as p.

Details

This is the single-lognormal counterpart of lognormal_mixgpd(). If \(F_{LN}(u)\) denotes the bulk probability below the threshold, then the spliced density is $$ f(x) = \left\{ \begin{array}{ll} f_{LN}(x \mid \mu,\sigma), & x < u, \\ \{1-F_{LN}(u)\} g_{GPD}(x \mid u,\sigma_u,\xi), & x \ge u. \end{array} \right. $$ The ordinary mean is finite only when the GPD tail has \(\xi < 1\); otherwise the package requires restricted means or quantiles.

Functions

• dLognormalGpd(): Lognormal + GPD tail density

• pLognormalGpd(): Lognormal + GPD tail distribution function

• rLognormalGpd(): Lognormal + GPD tail random generation

• qLognormalGpd(): Lognormal + GPD tail quantile function

See also

lognormal_mix(), lognormal_mixgpd(), gpd(), lognormal_lowercase().

Other lognormal kernel families: lognormal_mix, lognormal_mixgpd

Examples

meanlog <- 0.4
sdlog <- 0.35
threshold <- 3
tail_scale <- 0.9
tail_shape <- 0.2

dLognormalGpd(4.0, meanlog, sdlog, threshold, tail_scale, tail_shape, log = FALSE)
#> [1] 0.007654624
pLognormalGpd(4.0, meanlog, sdlog, threshold, tail_scale, tail_shape,
             lower.tail = TRUE, log.p = FALSE)
#> [1] 0.9915799
qLognormalGpd(0.50, meanlog, sdlog, threshold, tail_scale, tail_shape)
#> [1] 1.491825
qLognormalGpd(0.95, meanlog, sdlog, threshold, tail_scale, tail_shape)
#> [1] 2.65302
replicate(10, rLognormalGpd(1, meanlog, sdlog, threshold, tail_scale, tail_shape))
#>  [1] 1.129289 3.913359 2.131822 1.554833 1.452250 1.370077 1.738250 1.507340
#>  [9] 2.228381 3.814826