revdbayes (version 1.3.8)

kgaps_post: Random sampling from K-gaps posterior distribution

Description

Uses the rust package to simulate from the posterior distribution of the extremal index \(\theta\) based on the K-gaps model for threshold interexceedance times of Suveges and Davison (2010).

Usage

kgaps_post(
  data,
  thresh,
  k = 1,
  n = 1000,
  inc_cens = FALSE,
  alpha = 1,
  beta = 1,
  param = c("logit", "theta"),
  use_rcpp = TRUE
)

Arguments

data

A numeric vector of raw data. No missing values are allowed.

thresh

A numeric scalar. Extreme value threshold applied to data.

k

A numeric scalar. Run parameter \(K\), as defined in Suveges and Davison (2010). Threshold inter-exceedances times that are not larger than k units are assigned to the same cluster, resulting in a \(K\)-gap equal to zero. Specifically, the \(K\)-gap \(S\) corresponding to an inter-exceedance time of \(T\) is given by \(S = max(T - K, 0)\).

n

A numeric scalar. The size of posterior sample required.

inc_cens

A logical scalar indicating whether or not to include contributions from censored inter-exceedance times relating to the first and last observation. See Attalides (2015) for details.

alpha, beta

Positive numeric scalars. Parameters of a beta(\(\alpha\), \(\beta\)) prior for \(\theta\).

param

A character scalar. If param = "logit" (the default) then we simulate from the posterior distribution of \(\phi = \log(\theta / (1-\theta))\) and then transform back to the \(\theta\)-scale. If param = "theta" then we simulate directly from the posterior distribution of \(\theta\), unless the sample K-gaps are all equal to zero or all positive, when we revert to param = "logit". This is to avoid sampling directly from a posterior with mode equal to 0 or 1.

use_rcpp

A logical scalar. If TRUE (the default) the rust function ru_rcpp is used for posterior simulation. If FALSE the (slower) function ru is used.

Value

An object (list) of class "evpost", which has the same structure as an object of class "ru" returned from ru. In addition this list contains

• model: The character scalar "kgaps".

• thresh: The argument thresh.

• ss: The sufficient statistics for the K-gaps likelihood, as calculated by kgaps_stats.

Details

A beta(\(\alpha\), \(\beta\)) prior distribution is used for \(\theta\) so that the posterior from which values are simulated is proportional to $$\theta ^ {2 N_1 + \alpha - 1} (1 - \theta) ^ {N_0 + \beta - 1} \exp\{- \theta q (S_0 + \cdots + S_N)\}.$$ See kgaps_stats for a description of the variables involved in the contribution of the likelihood to this expression.

The ru function in the rust package simulates from this posterior distribution using the generalised ratio-of-uniforms distribution. To improve the probability of acceptance, and to ensure that the simulation will work even in extreme cases where the posterior density of \(\theta\) is unbounded as \(\theta\) approaches 0 or 1, we simulate from the posterior distribution of \(\phi = \log(\theta / (1-\theta))\) and then transform back to the \(\theta\)-scale.

References

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

Attalides, N. (2015) Threshold-based extreme value modelling, PhD thesis, University College London.

See Also

kgaps_mle for maximum likelihood estimation of the extremal index \(\theta\) using the K-gaps model.

kgaps_stats for the calculation of sufficient statistics for the K-gaps model.

ru for the form of the object returned by kgaps_post.

Examples

# NOT RUN {
thresh <- quantile(newlyn, probs = 0.90)
k_postsim <- kgaps_post(newlyn, thresh)
plot(k_postsim)
# }