The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance
AbstractEstimating 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 approaches. 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 in order 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 (2007). We show that the MLqE can outperform 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 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 traditionalMaximum Likelihood Estimator (MLE), recovering its desirable asymptotic properties; when q 6=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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Bibliographic InfoPaper provided by University of Modena and Reggio E., Faculty of Economics "Marco Biagi" in its series Department of Economics with number 555.
Length: pages 16
Date of creation: Jun 2007
Date of revision:
Maximum Likelihood; Extreme Value Theory; q-Entropy; Tail-related Risk Measures;
Other versions of this item:
- Davide Ferrari & Sandra Paterlini, 2007. "The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance," Center for Economic Research (RECent) 001, University of Modena and Reggio E., Dept. of Economics.
- Davide Ferrari & Sandra Paterlini, 2007. "The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 07071, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
- C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
- C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
- C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
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