thetaruns {tsxtreme}

Runs estimator

Description

Compute the empirical estimator of the extremal index using the runs method (Smith & Weissman, 1994, JRSSB).

Usage

thetaruns(ts, lapl = FALSE, nlag = 1,
          u.mar = 0, probs = seq(u.mar, 0.995, length.out = 30),
          method.mar = c("mle", "mom", "pwm"),
          R.boot = 0, block.length = (nlag+1) * 5, levels = c(0.025, 0.975))

Arguments

ts

a vector, the time series for which to estimate the threshold-based extremal index \theta(x,m), with x a probability level and m a run-length (see details).

lapl

logical; is ts on the Laplace scale already? The default (FALSE) assumes unknown marginal distribution.

nlag

the run-length; an integer larger or equal to 1.

u.mar

marginal threshold (probability); used when transforming the time series to Laplace scale if lapl is FALSE; if lapl is TRUE, it is nevertheless used when bootstrapping, since the bootstrapped series generally do not have Laplace marginal distributions.

probs

vector of probabilities; the values of x for which to evaluate \theta(x,m).

method.mar

a character string defining the method used to estimate the marginal GPD; either "mle" for maximum likelihood or "mom" for method of moments or "pwm" for probability weighted moments methods. Defaults to "mle".

block.length

integer; the block length used for the block-bootstrapped confidence intervals.

R.boot

integer; the number of samples used for the block bootstrap.

levels

vector of probabilites; the quantiles of the posterior distribution of the extremal index \theta(x,m) to output.

Details

Consider a stationary time series (X_t). A characterisation of the extremal index is

\theta(x,m) = Pr(X_1\le x,\ldots,X_m\le x \mid X_0\ge x).

In the limit when x and m tend to \infty appropriately, \theta corresponds to the asymptotic inverse mean cluster size. It also links the generalised extreme value distribution of the independent series (Y_t), with the same marginal distribution as (X_t),

G_Y(z)=G_X^\theta(z),

with G_X and G_Y the extreme value distributions of (X_t) and (Y_t) respectively.

nlag corresponds to the run-length m and probs is a set of values for x. The runs estimator is computed, which consists of counting the proportion of clusters to the number of exceedances of a threshold x; two exceedances of the threshold belong to different clusters if there are at least m+1 non-exceedances inbetween.

Value

An object of class 'depmeasure' containing:

theta

matrix; estimates of the extremal index \theta(x,m) with rows corresponding to the probs values of x and the columns to the runs estimate and the chosen levels-quantiles of the bootstrap distribution.

nbr.exc

numeric vector; number of exceedances for each threshold corresponding to the elements in probs.

probs

probs.

levels

numeric vector; probs converted to the original scale of ts.

nlag

nlag.

See Also

theta2fit, thetafit

Examples

## generate data from an AR(1)
## with Gaussian marginal distribution
n   <- 10000
dep <- 0.5
ar    <- numeric(n)
ar[1] <- rnorm(1)
for(i in 2:n)
  ar[i] <- rnorm(1, mean=dep*ar[i-1], sd=1-dep^2)
## transform to Laplace scale
ar <- qlapl(pnorm(ar))
## compute empirical estimate
theta <- thetaruns(ts=ar, u.mar=.95, probs=c(.95,.98,.99))
## output
plot(theta, ylim=c(.2,1))
abline(h=1, lty="dotted")