chibar: Parametric estimates of chi bar

In mev: Modelling of Extreme Values

Parametric estimates of \bar{\chi}

Description

The function fits a generalized Pareto distribution to minima of Pareto variates, using the representation

\Pr(\min(X) > x) = \frac{L(x)}{x^{1/\eta}},

where \bar{\chi}=2\eta-1. The data are transformed to the unit Pareto scale and a generalized Pareto variable is fitted to the minimum. The parameter \eta corresponds to the shape of the latter. The confidence intervals can be based either on the delta-method, a profile likelihood or a tangent exponential model approximation.

Usage

chibar(dat, confint = c("delta", "profile", "tem"), qu = 0, level = 0.95)

Arguments

dat	an n by d matrix of multivariate observations
confint	string indicating the type of confidence interval.
qu	percentile level at which to threshold. Default to all observations.
level	the confidence level required

Value

a named vector of length 3 containing the point estimate, the lower and the upper confidence intervals

See Also

chiplot for empirical estimates of \chi and \bar{\chi}.

Examples

## Not run: 
set.seed(765)
# Max-stable model, chibar = 1
dat <- rmev(n = 1000, model = "log", d = 2, param = 0.5)
chibar(dat, 'profile', qu = 0.5)
s <- seq(0.05,1, length = 30)
chibar_est <- t(sapply(s, function(keep){chibar(dat, 'delta', qu = keep)}))
matplot(s, chibar_est, type = 'l', col = c(1, 2, 2),  lty = c(1, 2, 2),
 ylab = expression(bar(chi)), xlab = 'p')
abline(h = 1, lty = 3, col = 'grey')
# Multivariate normal sample, chibar = 0 - strong asymptotic independence at penultimate level
dat <- mvrnorm(n = 1000, mu = c(0, 0), Sigma = cbind(c(1, 0.75), c(0.75, 1)))
chibar(dat, 'tem', q = 0.1)
chibar_est <- t(sapply(s, function(keep){chibar(dat, 'profile', qu = keep)}))
matplot(s, chibar_est, type = 'l', col = c(1, 2, 2),  lty = c(1, 2, 2),
 ylab = expression(bar(chi)), xlab = 'p')
abline(h = 1, lty = 3, col = 'grey')

## End(Not run)

mev documentation built on Sept. 11, 2024, 8:14 p.m.