Confidence intervals for the extremal index \(\theta\) for "kgaps" objects

confint method for objects of class c("kgaps", "exdex"). Computes confidence intervals for \(\theta\) based on an object returned from kgaps. Two types of interval may be returned: (a) intervals based on approximate large-sample normality of the estimator of \(\theta\), which are symmetric about the point estimate, and (b) likelihood-based intervals. The plot method plots the log-likelihood for \(\theta\), with the required confidence interval indicated on the plot.

# S3 method for kgaps
confint(
  object,
  parm = "theta",
  level = 0.95,
  interval_type = c("both", "norm", "lik"),
  conf_scale = c("theta", "log"),
  constrain = TRUE,
  se_type = c("observed", "expected"),
  ...
)

# S3 method for confint_kgaps
plot(x, ...)

# S3 method for confint_kgaps
print(x, ...)

Arguments

object

An object of class c("kgaps", "exdex"), returned by kgaps.

parm

Specifies which parameter is to be given a confidence interval. Here there is only one option: the extremal index \(\theta\).

level

The confidence level required. A numeric scalar in (0, 1).

interval_type

A character scalar: "norm" for intervals of type (a), "lik" for intervals of type (b).

conf_scale

A character scalar. If interval_type = "norm" then conf_scale determines the scale on which we use approximate large-sample normality of the estimator to estimate confidence intervals.

If conf_scale = "theta" then confidence intervals are estimated for \(\theta\) directly. If conf_scale = "log" then confidence intervals are first estimated for \(\log\theta\) and then transformed back to the \(\theta\)-scale.

constrain

A logical scalar. If constrain = TRUE then any confidence limits that are greater than 1 are set to 1, that is, they are constrained to lie in (0, 1]. Otherwise, limits that are greater than 1 may be obtained. If constrain = TRUE then any lower confidence limits that are less than 0 are set to 0.

se_type

A character scalar. Should the confidence intervals for the interval_type = "norm" use the estimated standard error based on the observed information or based on the expected information?

...

plot.confint_kgaps: further arguments passed to plot.confint.

print.confint_kgaps: further arguments passed to print.default.

x

an object of class c("confint_kgaps", "exdex"), a result of a call to confint.kgaps.

Value

A list of class c("confint_kgaps", "exdex") containing the following components.

cis

A matrix with columns giving the lower and upper confidence limits. These are labelled as (1 - level)/2 and 1 - (1 - level)/2 in % (by default 2.5% and 97.5%). The row names indicate the type of interval: norm for intervals based on large sample normality and lik for likelihood-based intervals. If object$k = 0 then both confidence limits are returned as being equal to the point estimate of \(\theta\).

call

The call to spm.

object

The input object object.

level

The input level.

plot.confint_kgaps: nothing is returned. If

x$object$k = 0 then no plot is produced.

print.confint_kgaps: the argument x, invisibly.

Details

Two type of interval are calculated: (a) an interval based on the approximate large sample normality of the estimator of \(\theta\) (if conf_scale = "theta") or of \(\log\theta\) (if conf_scale = "log") and (b) a likelihood-based interval, based on the approximate large sample chi-squared, with 1 degree of freedom, distribution of the log-likelihood ratio statistic.

print.confint_kgaps prints the matrix of confidence intervals for \(\theta\).

References

Suveges, M. and Davison, A. C. (2010) Model misspecification in peaks over threshold analysis, Annals of Applied Statistics, 4(1), 203-221. doi:10.1214/09-AOAS292

See also

kgaps for estimation of the extremal index \(\theta\) using a semiparametric maxima method.

Examples

u <- quantile(newlyn, probs = 0.90)
theta <- kgaps(newlyn, u)
cis <- confint(theta)
cis
#>          2.5 %    97.5 %
#> norm 0.3334645 0.4223252
#> lik  0.3345565 0.4233068
plot(cis)