R: Maximum likelihood estimation for the K-gaps model

kgaps {exdex}

R Documentation

Maximum likelihood estimation for the `K`-gaps model

Description

Calculates maximum likelihood estimates of the extremal index \theta based on the K-gaps model for threshold inter-exceedances times of Suveges and Davison (2010).

Usage

kgaps(data, u, k = 1, inc_cens = TRUE)

Arguments

`data`	A numeric vector or numeric matrix of raw data. If `data` is a matrix then the log-likelihood is constructed as the sum of (independent) contributions from different columns. A common situation is where each column relates to a different year. If `data` contains missing values then `split_by_NAs` is used to divide the data further into sequences of non-missing values, stored in different columns in a matrix. Again, the log-likelihood is constructed as a sum of contributions from different columns.
`u`	A numeric scalar. Extreme value threshold applied to data.
`k`	A non-negative 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)`. In practice, `k` should be no smaller than 1, because when `k` is less than 1 the estimate of `\theta` is always equal to 1.
`inc_cens`	A logical scalar indicating whether or not to include contributions from right-censored inter-exceedance times, relating to the first and last observations. It is known that these times are greater than or equal to the time observed. See Attalides (2015) for details. If `data` has multiple columns then there will be right-censored first and last inter-exceedance times for each column.

Details

If inc_cens = FALSE then the maximum likelihood estimate of the extremal index \theta under the K-gaps model of Suveges and Davison (2010) is calculated.

If inc_cens = TRUE then information from right-censored first and last inter-exceedance times is also included in the likelihood to be maximized, following Attalides (2015). The form of the log-likelihood is given in the Details section of kgaps_stat.

It is possible that the estimate of \theta is equal to 1, and also possible that it is equal to 0. kgaps_stat explains the respective properties of the data that cause these events to occur.

Value

An object (a list) of class c("kgaps", "exdex") containing

`theta`	The maximum likelihood estimate (MLE) of `\theta`.
`se`	The estimated standard error of the MLE, calculated using an algebraic expression for the observed information. If `k = 0` then `se` is returned as `0`.
`se_exp`	The estimated standard error of the MLE, calculated using an algebraic expression for the expected information. If the estimate of `\theta` is 0 or 1 then `se_exp` is `NA`.
`ss`	The list of summary statistics returned from `kgaps_stat`.
`k`, `u`, `inc_cens`	The input values of `k`, `u` and `inc_cens`.
`max_loglik`	The value of the log-likelihood at the MLE.
`call`	The call to `kgaps`.

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

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

Examples

### S&P 500 index

u <- quantile(sp500, probs = 0.60)
theta <- kgaps(sp500, u)
theta
summary(theta)
coef(theta)
nobs(theta)
vcov(theta)
logLik(theta)

### Newlyn sea surges

u <- quantile(newlyn, probs = 0.60)
theta <- kgaps(newlyn, u, k = 2)
theta
summary(theta)

### Cheeseboro wind gusts

theta <- kgaps(cheeseboro, 45, k = 3)
theta
summary(theta)

[Package exdex version 1.2.3 Index]

Maximum likelihood estimation for the K-gaps model