mev (version 1.17)

chibar: 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)

Value

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

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

See Also

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

Examples

if (FALSE) {
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')
}