The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance

Davide Ferrari and Sandra Paterlini

Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) from Universita di Modena e Reggio Emilia, Dipartimento di Economia "Marco Biagi"

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 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 traditional Maximum 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).

Keywords: Maximum Likelihood; Extreme Value Theory; q-Entropy; Tail-related risk measures (search for similar items in EconPapers)
JEL-codes: C13 C22 C51 (search for similar items in EconPapers)
Pages: pages 20
Date: 2007-07
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (7)

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Related works:
Journal Article: The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance (2009)
Working Paper: The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance (2007)
Working Paper: The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance (2007)
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