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Correction to: Automatic threshold and run parameter selection: a climatology for extreme hourly precipitation in Switzerland

  • S. FukutomeEmail author
  • M. A. Liniger
  • M. Süveges
Correction
  • 52 Downloads

Correction to: Theor Appl Climatol (2015) 120:403–416

 https://doi.org/10.1007/s00704-014-1180-5

The original version of this article unfortunately contained a mistake. Please find below the relevant part of the Appendix, with changes in the last 2 formulae.

Let i, ji, ii, di, denote for a single observation i: the log-likelihood, the score function, the expected information, and the difference between score function and expected information, respectively. Let the derivative with respect to θ be denoted by a prime. Let \( \mathbb{I}(A) \) be the indicator function for the set A. Then, for a given (u, K) pair,
$$ {\ell}_i^{\hbox{'}}\left(\theta \right)=-\frac{\mathbb{I}\left({c}_i^{\left(u,K\right)}=0\right)}{\left(1-\theta \right)}+\frac{2\mathbb{I}\left({c}_i^{\left(u,K\right)}>0\right)}{\theta }-{c}_i^{\left(u,K\right)}, $$
$$ {j}_i\left(\theta \right)=\frac{\mathbb{I}\left({c}_i^{\left(u,K\right)}=0\right)}{{\left(1-\theta \right)}^2}+\frac{4\mathbb{I}\left({c}_i^{\left(u,K\right)}>0\right)}{\theta^2}+{\left({c}_i^{\left(u,K\right)}\right)}^2-\frac{4{c}_i^{\left(u,K\right)}}{\theta }, $$
$$ {i}_i\left(\theta \right)=\frac{\mathbb{I}\left({c}_i^{\left(u,K\right)}=0\right)}{{\left(1-\theta \right)}^2}+\frac{2\mathbb{I}\left({c}_i^{\left(u,K\right)}>0\right)}{\theta^2}, $$
$$ {d}_i\left(\theta \right)=\frac{2\mathbb{I}\left({c}_i^{\left(u,K\right)}>0\right)}{\theta^2}+{\left({c}_i^{\left(u,K\right)}\right)}^2-\frac{4{c}_i^{\left(u,K\right)}}{\theta }, $$
$$ {d_i}^{\hbox{'}}\left(\theta \right)=-\frac{4\mathbb{I}\left({c}_i^{\left(u,K\right)}>0\right)}{\theta^3}+\frac{4{c}_i^{\left(u,K\right)}}{\theta^2}. $$
Let \( D\left(\theta \right)={\left(N-1\right)}^{-1}\sum \limits_{k=1}^{N-1}{d}_k\left(\theta \right) \) and \( I\left(\theta \right)={\left(N-1\right)}^{-1}\sum \limits_{k=1}^{N-1}{i}_k\left(\theta \right) \) denote the sample means of di and ii. The sample variance of D(θ) is:
$$ V\left(\theta \right)={\left(N-1\right)}^{-1}\sum \limits_{k=1}^{N-1}\left\{{\left[{d}_k\left(\theta \right)-{D}^{\hbox{'}}\left(\theta \right)I{\left(\theta \right)}^{-1}{\ell_k}^{\hbox{'}}\left(\theta \right)\right]}^2\right\}. $$
The Information Matrix Test (IMT) Statistic is then:
$$ IMT\left(\widehat{\theta}\right)=\left(N-1\right)D{\left(\widehat{\theta}\right)}^2V{\left(\widehat{\theta}\right)}^{-1} $$
where θ has been replaced by the estimated value of \( \widehat{\theta} \).

Notes

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Federal Office of Meteorology and ClimatologyZürich-FlughafenSwitzerland
  2. 2.Department of Astronomy, ISDC Data Centre for AstrophysicsUniversity of GenevaVersoixSwitzerland

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