# kgaps_post: Random sampling from K-gaps posterior distribution In revdbayes: Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis

## Description

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

## Usage

 ```1 2``` ```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(α, β) prior for θ. `param` A character scalar. If `param = "logit"` (the default) then we simulate from the posterior distribution of φ = logit(θ) and then transform back to the θ-scale. If `param = "theta"` then we simulate directly from the posterior distribution of θ, 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.

## Details

A beta(α, β) prior distribution is used for θ so that the posterior from which values are simulated is proportional to

θ ^ (2 N_1 + α - 1) * (1 - θ) ^ (N_0 + β - 1) * exp(- θ q (S_0 + ... + 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 θ is unbounded as θ approaches 0 or 1, we simulate from the posterior distribution of φ = logit(θ) and then transform back to the θ-scale.

## 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`.

## References

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

Attalides, N. (2015) Threshold-based extreme value modelling, PhD thesis, University College London. http://discovery.ucl.ac.uk/1471121/1/Nicolas_Attalides_Thesis.pdf

`kgaps_mle` for maximum likelihood estimation of the extremal index θ 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

 ```1 2 3``` ```thresh <- quantile(newlyn, probs = 0.90) k_postsim <- kgaps_post(newlyn, thresh) plot(k_postsim) ```

revdbayes documentation built on Feb. 13, 2018, 1:04 a.m.