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Density, Distribution Function, Quantile Function and Random Generation for the Five-Parameter Compound Generalized Pareto Distribution (CGPD)

CGPD.Rd

Density, distribution function, quantile function and random generation for the five-parameter Compound Generalized Pareto Distibution (CGPD).

Usage

dCGPD(x, loc = 0.0, scale = 1.0, shape = 0.0,
      scaleN, shapeN, EN, IDN, log = FALSE)

pCGPD(q, loc = 0.0, scale = 1.0, shape = 0.0,
      scaleN, shapeN, EN, IDN, lower.tail = TRUE)

qCGPD(p, loc = 0.0, scale = 1.0, shape = 0.0,
      scaleN, shapeN, EN, IDN, lower.tail = TRUE)

rCGPD(n, loc = 0.0, scale = 1.0, shape = 0.0,
      scaleN, shapeN, EN, IDN)

pCGPD(
  q,
  loc = 0,
  scale = 1,
  shape = 0,
  scaleN,
  shapeN,
  EN,
  IDN,
  lower.tail = TRUE
)

qCGPD(
  p,
  loc = 0,
  scale = 1,
  shape = 0,
  scaleN,
  shapeN,
  EN,
  IDN,
  lower.tail = TRUE
)

rCGPD(n, loc = 0, scale = 1, shape = 0, scaleN, shapeN, EN, IDN)

Arguments

x, q

Vector of quantiles.

loc

Location parameter. Numeric vector of length one.

scale

Scale parameter. Numeric vector of length one.

shape

Shape parameter. Numeric vector of length one.

scaleN

Scale of the GPD for the \(N\) part. Along with shapeN it provides the parameterisation for the Binomial-Poisson-Negative Binomial familly.

shapeN

Shape of the GPD for the \(N\) part. Along with scaleN it provides the parameterisation for the Binomial-Poisson-Negative Binomial familly.

EN

Expectation of \(N\). Along with IDN it provides an alternative parameterisation for the \(N\) part.

IDN

Index of Dispersion of \(N\). Along with EN it provides an alternative parameterisation for the \(N\) part.

log

Logical; if TRUE, densities p are returned as log(p).

lower.tail

Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

p

Vector of probabilities.

n

Sample size.

Value

A numeric vector with length equal to the length of the first argument or of the parameters.

Details

This distribution is that of the maximum \(M\) of \(N\) i.i.d. r.vs \(X_i\) with distribution \(\textrm{GPD}(\mu,\,\sigma,\,\xi)\) where \(N\) is a r.v. with non-negative integer values, independent of the sequence \(X_i\), and having a Binomial, Poisson or Negative Binomial distribution. The distribution of \(N\) can be parameterized by using two parameters \(\mu_N\) and \(\sigma_N\) in a GPD style, or alternatively by using the two parameters \(\mathrm{E}(N)\) and \(\textrm{ID}(N)\) representing the expectation and the index of dispersion of \(N\). The three cases Binomial, Poisson and Negative Binomial correspond to \(\textrm{ID}_N < 0\), \(\textrm{ID}_N = 0\) and \(\textrm{ID}_N > 0\).

Caution

This distribution is of mixed-type. It has a probability mass at \(-\infty\) corresponding to the possibility that \(N=0\) in which case \(M\) is the maximum of an empty set, taken as \(-\infty\) corresponding to max(mumeric(0)). Consequently a sample drawn by using rCGPD contains -Inf values with positive probability.

References

Yves Deville (2019) "Bayesian Return Levels in Extreme-Value Analysis" IRSN technical report.

Author

Yves Deville

Examples

set.seed(1)
ExpN <- runif(1)
IDN <- rexp(1, rate = 1)
scaleN <- 1 / ExpN
shapeN <- (IDN - 1) / ExpN
loc <- rnorm(1, mean = 0, sd = 10); scale <-  rexp(1)
shape <- rnorm(1 , mean = 0, sd = 0.1)
mass <- pCGPD(-Inf, scaleN = scaleN, shapeN = shapeN,
           loc = loc, scale = scale, shape = shape) 
q <- qCGPD(p = c(mass + 0.001, 0.999), scaleN = scaleN, shapeN = shapeN,
           loc = loc, scale = scale, shape = shape)
x <- seq(from = q[1] - 1, to = q[2], length.out = 200)
F <- pCGPD(x, scaleN = scaleN, shapeN = shapeN,
           loc = loc, scale = scale, shape = shape)
plot(x, F, type = "l", xlab = "", ylab = "", ylim = c(0, 1),
     col = "orangered")
abline(h = mass, col = "red")

f <- dCGPD(x, scaleN = scaleN, shapeN = shapeN,
           loc = loc, scale = scale, shape = shape)
plot(x, f, type = "l", col = "SteelBlue3", xlab = "", ylab = "")
title(main = sprintf(paste("ExpN = %4.1f IDN = %4.2f,",
                           "loc = %4.1f, scale = %4.2f, shape = %4.2f"),
                     ExpN, IDN, loc, scale, shape))