evgam (version 1.0.2)

qev: Quantile estimation of a composite extreme value distribution

Description

Quantile estimation of a composite extreme value distribution

Usage

qev(
  p,
  loc,
  scale,
  shape,
  m = 1,
  alpha = 1,
  theta = 1,
  family,
  tau = 0,
  bgev.args = list(pa = 0.05, pb = 0.2, alpha = 0.5, beta = 0.5),
  start = NULL,
  method = "uniroot"
)

Value

A scalar or vector of estimates of p

Arguments

p

a scalar giving the quantile of the distribution sought

loc

a scalar, vector or matrix giving the location parameter

scale

as above, but scale parameter

shape

as above, but shape parameter

m

a scalar giving the number of values per return period unit, e.g. 365 for daily data giving annual return levels

alpha

a scalar, vector or matrix of weights if within-block variables not identically distributed and of different frequencies

theta

a scalar, vector or matrix of extremal index values

family

a character string giving the family for which return levels sought

tau

a scalar, vector or matrix of values giving the threshold quantile for the GPD (i.e. 1 - probability of exceedance)

bgev.args

a list specifying parameters of the blended GEV distribution; see Details

start

a 2-vector giving starting values that bound the return level

method

a character string giving the numerical estimation procedure; defaults to "uniroot"

Details

If \(F\) is the generalised extreme value, generalised Pareto or blended GEV distribution, qev solves $$\prod_{j=1}^n \big\{F_i(z)\}^{m \alpha_j \theta_j} = p.$$ for \(i = 1, \ldots, k\). So vectors are supplied as $n$-vectors and matrices are supplied as \(n \times k\) matrices.

For all distributions, location, scale and shape parameters are given by loc, scale and shape. The generalised Pareto distribution, for \(\xi \neq 0\) and \(z > u\), is parameterised as \(1 - (1 - \tau) [1 + \xi (z - u) / \psi_u]^{-1/\xi}\), where \(u\), \(\psi_u\) and \(\xi\) are its location, scale and shape parameters, respectively, and \(\tau\) corresponds to argument tau. For the blended GEV distribution pa, pb, alpha and beta specify additional parameters of the blended GEV distribution; see family.evgam for details.

Estimates either use function uniroot or method = "newton" uses the Newton-Rhaphson method. The latter is often much quicker if matrices are supplied, i.e. for \(k > 1\).

Examples

qev(0.9, c(1, 2), c(1, 1.1), .1, family = "gev")
qev(0.99, c(1, 2), c(1, 1.1), .1, family = "gpd", tau = 0.9)

# an example representative on monthly estimates at two locations
qev(0.9, matrix(c(1:12, 2:13), 12, 2), 1.1, .1, family = "gev")

# a blended GEV example with default blended GEV specification
qev(0.9, matrix(c(1:12, 2:13), 12, 2), 1.1, .1, family = "bgev")