The Maximum Lq-likelihood method: an application to extreme quantile estimation in finance

Estimating financial risk is a critical issue for banks and insurance companies.Recently, quantile estimation based on extreme value theory (EVT) has founda successful domain of application in such a context, outperforming other methods.Given a parametric model provided by EVT, a natural approach is maximumlikelihood estimation. Although the resulting estimator is asymptotically efficient,often the number of observations available to estimate the parameters of the EVTmodels 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 themaximum Lq-likelihood method, Technical Report 659, School of Statistics, Universityof Minnesota, 2007 http://www.stat.umn.edu/~dferrari/research/techrep659.pdf).We show that the MLqE outperforms the standard MLE, when estimating tail probabilitiesand quantiles of the generalized extreme value (GEV) and the generalizedPareto (GP) distributions. First, we assess the relative efficiency between the MLqEand the MLE for various sample sizes, using Monte Carlo simulations. Second, weanalyze the performance of the MLqE for extreme quantile estimation using realworldfinancial data. The MLqE is characterized by a distortion parameter q andextends the traditional log-likelihood maximization procedure. When q different from 1, thenew estimator approaches the traditional maximum likelihood estimator (MLE),recovering its desirable asymptotic properties; when q = 1 and the sample size ismoderate or small, the MLqE successfully trades bias for variance, resulting in anoverall gain in terms of accuracy (mean squared error).