Statistics for the information matrix test

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 kgaps.

u

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 $S = \max(T - K, 0)$.

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.

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 $K$-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_kgaps

the number of $K$-gaps that contribute to the log-likelihood.

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