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Amoroso with a GPD tail

amoroso_gpd.Rd

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

Usage

dAmorosoGpd(
  x,
  loc,
  scale,
  shape1,
  shape2,
  threshold,
  tail_scale,
  tail_shape,
  log = 0
)

pAmorosoGpd(
  q,
  loc,
  scale,
  shape1,
  shape2,
  threshold,
  tail_scale,
  tail_shape,
  lower.tail = 1,
  log.p = 0
)

rAmorosoGpd(n, loc, scale, shape1, shape2, threshold, tail_scale, tail_shape)

qAmorosoGpd(
  p,
  loc,
  scale,
  shape1,
  shape2,
  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.

loc

Numeric scalar location parameter of the Amoroso bulk.

scale

Numeric scalar scale parameter of the Amoroso bulk.

shape1

Numeric scalar first Amoroso shape parameter.

shape2

Numeric scalar second Amoroso shape parameter.

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

Logical; if TRUE, return the log-density (integer flag 0/1 in NIMBLE).

q

Numeric scalar giving the point at which the distribution function is evaluated.

lower.tail

Logical; if TRUE (default), probabilities are \(P(X \le q)\).

log.p

Logical; if TRUE, 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 tolerance for numerical inversion in qAmorosoGpd.

maxiter

Maximum iterations for numerical inversion in qAmorosoGpd.

Value

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

Details

This is the single-component Amoroso splice. The Amoroso law controls the distribution below the threshold and the GPD controls exceedances above it, scaled so that the resulting CDF is continuous at the threshold. The ordinary mean of the spliced law exists only when the tail satisfies \(\xi < 1\); otherwise users should rely on restricted means or quantile summaries.

Functions

  • dAmorosoGpd(): Density Function of Amoroso Distribution with GPD Tail

  • pAmorosoGpd(): Cumulative Distribution Function of Amoroso Distribution with GPD Tail

  • rAmorosoGpd(): Random Generation for Amoroso Distribution with GPD Tail

  • qAmorosoGpd(): Quantile Function of Amoroso Distribution with GPD Tail

See also

amoroso_mix(), amoroso_mixgpd(), gpd(), amoroso_lowercase().

Other amoroso kernel families: amoroso_mix, amoroso_mixgpd

Examples

loc <- 0
scale <- 1.5
shape1 <- 2
shape2 <- 1.2
threshold <- 3
tail_scale <- 1.0
tail_shape <- 0.2

dAmorosoGpd(4.0, loc, scale, shape1, shape2,
           threshold, tail_scale, tail_shape, log = 0)
#> [1] 0.1110036
pAmorosoGpd(4.0, loc, scale, shape1, shape2,
           threshold, tail_scale, tail_shape, lower.tail = 1, log.p = 0)
#> [1] 0.8667957
qAmorosoGpd(0.50, loc, scale, shape1, shape2,
           threshold, tail_scale, tail_shape)
#> [1] 2.309366
qAmorosoGpd(0.95, loc, scale, shape1, shape2,
           threshold, tail_scale, tail_shape)
#> [1] 5.298959
replicate(10, rAmorosoGpd(1, loc, scale, shape1, shape2,
                         threshold, tail_scale, tail_shape))
#>  [1] 1.2615281 0.7103682 6.3013993 3.0468416 3.2229290 4.5536784 2.4216707
#>  [8] 4.3138464 1.3671128 2.8536680