Abstract
Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 http//:www.stat.umn.edu/~dferrari/research/techrep659.pdf ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).
Highlights
Recent financial crises and the new regulations for banks and insurance companies1 have prompted intermediaries to regularly compute statistical tail-related measures of risk
The POT method exploits the Generalized Pareto (GP) distribution for modelling the exceedances over a certain threshold, while the Block Maxima (BM) method relies on the Generalized Extreme Value (GEV) distribution to model the maximum value that a variable takes in a given period of time
We have shown that the Maximum Lq-Likelihood Estimator (MLqE) can be a valid alternative to the classical Maximum Likelihood Estimator (MLE) when estimating a small tail probability or a large quantile in the context of Extreme Value Theory
Summary
Recent financial crises and the new regulations for banks and insurance companies have prompted intermediaries to regularly compute statistical tail-related measures of risk. Our investigation aims to address this issue by studying for the first time in the EVT context the performance of a new estimator of the parameters, the Maximum Lq-Likelihood Estimator (MLqE), which has been recently proposed by Ferrari and Yang (2007). The MLqE is based on the information measure introduced by Havrda and Charvat (1967) and generalizes the traditional log-likelihood maximization procedure: it preserves the desirable asymptotic properties of the traditional MLE, while it allows for a peculiar type of distortion introduced by the extra parameter q, resulting in a gain in terms of precision (Mean Squared Error) when the sample size is moderate or small. The objective of this paper is to study the behavior of the new estimator on both simulated data and on real-world time series for extreme quantile estimation.
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