Implementation of the paper "A refined Weissman estimator of extreme quantile", by Michaël Allouche, Jonathan El Methni and Stéphane Girard.
The repo contains the codes for comparing our proposed extreme quantile estimator with 7 other known estimators in the literature on both simulated and real-data.
Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distribution is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. This new bias reduction method is fully automatic and does not involve the selection of extra parameters. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms these competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided. Our approach is compared to other bias reduced estimators of extreme quantiles both on simulated and real data.
Clone the repo
git clone https://github.com/michael-allouche/refined-weissman.git
cd refined-weissman
Install the requirements for each software version used in this repo
- Python 3.10.1
via pip
pip install -r requirements.txt
or via conda
conda create --name rweissman python=3.10
conda activate rweissman
conda install --file requirements.txt
- R 4.1.2
install.packages("evt0")
Seven heavy-tailed distributions are implemented in ./extreme/distribution.py:
Burr, NHW, Fréchet, Fisher, GPD, Inverse Gamma, Student.
In run_evt_estimators.py, one can update the dict_runner with the desired parametrization.
Next, run run_evt_estimators.py to compute all the quantile estimators at both quantile levels alpha=1/n and alpha=1/(2n) .
For example, estimations applied to 1000 replications of 500 samples issued from a Burr distribution:
python run_evt_estimators.py -d burr -r 1000 -n 500
Once the run is finished, all the metrics for each estimator are saved in the folder ./ckpt.
In the notebook, you can display a result table. For example
from extreme.estimators import evt_estimators
evt_estimators(n_replications=1000, params={"evi":0.125, "rho": -1.},
distribution="burr",
n_data=500, n_quantile="2n")
Estimators W RW CW CH CHp PRBp CHps PRBps
RMSE 0.0471 0.0095 0.0063 0.0155 0.0149 0.015 0.0135 0.0164
You can also plot the bias, the variance and the RMSE
from extreme import visualization as statsviz
statsviz.evt_quantile_plot(n_replications=1000,
params={"evi":0.125, "rho": -0.125},
distribution="burr",
n_data=500,
n_quantile="2n")
We consider here the Secura Belgian reinsurance data set in ./dataset/besecura.txt on automobile claims from 1998 until 2001.
This data set consists of claims which were at least as large as 1.2 million Euros and were corrected for inflation.
Our goal is to estimate the extreme quantile
(with
) and to compare it to the maximum
of the sample
million Euros.
Below the histogram of the losses:
Display the quantile plot and the selected intemediate sequence
using our proposed Algorithm in the paper.
statsviz.real_quantile_plot()
@article{allouche2023refined, title={A refined Weissman estimator for extreme quantiles}, author={Allouche, Micha{"e}l and El Methni, Jonathan and Girard, St{'e}phane}, journal={Extremes}, volume={26}, number={3}, pages={545--572}, year={2023}, publisher={Springer} }