Efficient Density Estimation and Value at Risk Using Fejér-Type Kernel Functions


This paper presents a nonparametric method for computing the Value at Risk (VaR) based on efficient density estimators with Fejér-type kernel functions and empirical bandwidths obtained from Fourier analysis techniques. The kernel-type estimator with a Fejér-type kernel was recently found to dominate all other known density estimators under the -risk, . This theoretical finding is supported via simulations by comparing the quality of the density estimator in question with other fixed kernel estimators using the common -risk. Two data-driven bandwidth selection methods, cross-validation and the one based on the Fourier analysis of a kernel density estimator, are used and compared to the theoretical bandwidth. The proposed nonparametric method for computing the VaR is applied to two fictitious portfolios. The performance of the new VaR computation method is compared to the commonly used Gaussian and historical simulation methods using a standard back-test procedure. The obtained results show that the proposed VaR model provides more reliable estimates than the standard VaR models.

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Kosta, O. and Stepanova, N. (2015) Efficient Density Estimation and Value at Risk Using Fejér-Type Kernel Functions. Journal of Mathematical Finance, 5, 480-504. doi: 10.4236/jmf.2015.55040.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Jorion, P. (2001) Value at Risk: The New Benchmark for Managing Financial Risk. 2nd Edition, McGraw-Hill, United States of America
[2] Stepanova, N. (2013) On Estimation of Analytic Density Functions in Lp. Mathematical Methods of Statistics, 22, 114-136.
[3] Levit, B. and Stepanova, N. (2004) Efficient Estimation of Multivariate Analytic Functions in Cube-Like Domains. Mathematical Methods of Statistics, 13, 253-281.
[4] Golubev, G.K., Levit, B.Y. and Tsybakov, A.B. (1996) Asymptotically Efficient Estimation of Analytic Functions in Gaussian Noise. Bernoulli, 2, 167-181.
[5] Guerre, E. and Tsybakov, A.B. (1998) Exact Asymptotic Minimax Constants for the Estimation of Analytic Functions in Lp. Probability Theory and Related Fields, 112, 33-51.
[6] Schipper, M. (1996) Optimal Rates and Constants in L2-Minimax Estimation. Mathematical Methods of Statistics, 5, 253-274.
[7] Hájek, J. (1972) Local Asymptotic Minimax and Admissibility in Estimation. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 175-194.
[8] Belitser, E. (1998) Efficient Estimation of Analytic Density under Random Censorship. Bernoulli, 4, 519-543.
[9] Ibragimov, I.A. and Hasminskii, R.Z. (1983) On Estimation of the Density Function. Journal of Soviet Mathematics, 25, 40-57.
[10] Rudin, W. (1987) Real and Complex Analysis. 3rd Edition, McGraw-Hill, Singapore.
[11] Tsybakov, A.B. (2009) Introduction to Nonparametric Estimation. Springer Science, United States of America.
[12] Kosta, O. (2015) Efficient Density Estimation Using Fejér-Type Kernel Functions. M.Sc. Thesis, Carleton University, Ottawa.
[13] Cline, D.B.H. (1988) Admissible Kernel Estimators of a Multivariate Density. Annals of Statistics, 16, 1421-1427.
[14] Härdle, W., Müller, M., Sperlich, S. and Weratz, A. (2004) Nonparametric and Semiparametric Models. Springer, Heidelberg.
[15] Rudemo, M. (1982) Empirical Choice of Histograms and Kernel Density Estimators. Scandinavian Journal of Statistics, 9, 65-78.
[16] Scott, D.W. and Terrell, G.R. (1987) Biased and Unbiased Cross-Validation in Density Estimation. Journal of the American Statistical Association, 82, 1131-1146.
[17] Golubev, G.K. (1992) Nonparametric Estimation of Smooth Densities of a Distribution in L2. Problems of Information Transmission, 23, 57-67.
[18] Achieser, N.I. (1956) Theory of Approximation. Frederick Ungar Publishing, New York.
[19] Golubev, G.K. and Levit, B.Y. (1996) Asymptotically Efficient Estimation for Analytic Distributions. Mathematical Methods of Statistics, 5, 357-368.
[20] Mason, D.M. (2010) Risk Bounds for Kernel Density Estimators. Journal of Mathematical Sciences, 163, 238-261.
[21] Lepski, O.V. and Levit, B.Y. (1998) Adaptive Minimax Estimation of Infinitely Differentiable Functions. Mathematical Methods of Statistics, 7, 123-156.
[22] Parzen, E. (1979) Nonparametric Statistical Data Modeling. Journal of the American Statistical Association, 74, 105-121.
[23] Marron, J.S. and Sheather, S.J. (1990) Kernel Quantile Estimators. Journal of the American Statistical Association, 85, 410-416.
[24] Roussas, G. (1997) A Course in Mathematical Statistics. 2nd Edition, Academic Press, United States of America.

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