The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance |
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Authors: | Davide Ferrari Sandra Paterlini |
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Institution: | (1) School of Statistics, University of Minnesota, 313 Ford Hall, 224 Church Street S.E., Minneapolis, MN 55455, USA;(2) Department of Economics, CEFIN—Research Center on Banking and Finance, RECent—Center for Economic Research, University of Modena and Reggio Emilia, Viale J. Berengario, 51, 41100 Modena, Italy |
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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
). 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).
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Keywords: | Maximum likelihood Extreme value theory q-Entropy Tail-related risk measures |
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