empfit: Runs estimator
In tsxtreme: Bayesian Modelling of Extremal Dependence in Time Series

Description Usage Arguments Details Value See Also Examples

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

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))

`ts`	a vector, the time series for which to estimate the threshold-based extremal index θ(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 θ(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 θ(x,m) to output.

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

θ(x,m) = Pr(X_1≤ x,…,X_m≤ x | X_0≥ x).

In the limit when x and m tend to ∞ appropriately, θ 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^θ(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.

An object of class 'depmeasure' containing:

`theta`	matrix; estimates of the extremal index θ(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`.

theta2fit, thetafit

## 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")