Statistics for the \(D\)-gaps information matrix test

Calculates the components required to calculate the value of the information matrix test under the \(D\)-gaps model, using vector data input. Called by dgaps_imt.

Usage

dgaps_imt_stat(data, theta, u, D = 1, inc_cens = TRUE)

Arguments

data

A numeric vector of raw data. Missing values are allowed, but they should not appear between non-missing values, that is, they only be located at the start and end of the vector. Missing values are omitted using na.omit.

theta

A numeric scalar. An estimate of the extremal index \(\theta\), produced by dgaps.

u

A numeric scalar. Extreme value threshold applied to data.

D

A numeric scalar. The censoring parameter \(D\). Threshold inter-exceedances times that are not larger than D units are left-censored, occurring with probability \(\log(1 - \theta e^{-\theta d})\), where \(d = q D\) and \(q\) is the probability with which the threshold \(u\) is exceeded.

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. See dgaps.

Value

A list relating the quantities given on pages 18-19 of Suveges and Davison (2010). All but the last component are vectors giving the contribution to the quantity from each \(D\)-gap, evaluated at the input value theta of \(\theta\).

ldj

the derivative of the log-likelihood with respect to \(\theta\) (the score)

Ij

the observed information

Jj

the square of the score

dj

Jj - Ij

Ddj

the derivative of Jj - Ij with respect to \(\theta\)

n_dgaps

the number of \(D\)-gaps that contribute to the log-likelihood.

References

Holesovsky, J. and Fusek, M. Estimation of the extremal index using censored distributions. Extremes 23, 197-213 (2020). doi:10.1007/s10687-020-00374-3