Threshold \(u\) and runs parameter \(K\) diagnostic for the \(K\)-gaps estimator

Creates data for a plot to aid the choice of the threshold and run parameter \(K\) for the \(K\)-gaps estimator (see kgaps). plot.choose_uk creates the plot.

choose_uk(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 into sequences of non-missing values.

u, k

Numeric vectors. u is a vector of extreme value thresholds applied to data. k is a vector of values of the run parameter \(K\), as defined in Suveges and Davison (2010). See kgaps for more details.

Any values in u that are greater than all the observations in data will be removed without a warning being given.

inc_cens

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

Value

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

containing

imt

an object of class c("kgaps_imt", "exdex") returned from kgaps_imt.

theta

a length(u) by length(k) matrix. Element (i,j) of theta contains an object (a list) of class c("kgaps", "exdex"), a result of a call kgaps(data, u[j], k[i]) to kgaps.

Details

For each combination of threshold in u and \(K\) in k the functions kgaps and kgaps_imt are called in order to estimate \(\theta\) and to perform the information matrix test of Suveges and Davison (2010).

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

See also

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

kgaps_imt for the information matrix test under the \(K\)-gaps model

plot.choose_uk to produce the diagnostic plot.

Examples

### S&P 500 index

# Multiple thresholds and run parameters
u <- quantile(sp500, probs = seq(0.1, 0.9, by = 0.1))
imt_theta <- choose_uk(sp500, u = u, k = 1:5)
plot(imt_theta)

plot(imt_theta, uprob = TRUE)

plot(imt_theta, y = "theta")


# One run parameter K, many thresholds u
u <- quantile(sp500, probs = seq(0.1, 0.9, by = 0.1))
imt_theta <- choose_uk(sp500, u = u, k = 1)
plot(imt_theta)

plot(imt_theta, y = "theta")


# One threshold u, many run parameters K
u <- quantile(sp500, probs = 0.9)
imt_theta <- choose_uk(sp500, u = u, k = 1:5)
plot(imt_theta)

plot(imt_theta, y = "theta")


### Newlyn sea surges

u <- quantile(newlyn, probs = seq(0.1, 0.9, by = 0.1))
imt_theta <- choose_uk(newlyn, u = u, k = 1:5)
plot(imt_theta, uprob = TRUE)


### Cheeseboro wind gusts (a matrix containing some NAs)

probs <- c(seq(0.5, 0.95, by = 0.05), 0.99)
u <- quantile(cheeseboro, probs = probs, na.rm = TRUE)
imt_theta <- choose_uk(cheeseboro, u, k = 1:6)
plot(imt_theta, uprob = FALSE, lwd = 2)


### Uccle July temperatures

probs <- c(seq(0.7, 0.95, by = 0.05), 0.99)
u <- quantile(uccle720m, probs = probs, na.rm = TRUE)
imt_theta <- choose_uk(uccle720m, u, k = 1:5)
plot(imt_theta, uprob = TRUE, lwd = 2)