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NONPARAMETRIC DENSITY ESTIMATION BY B-SPLINE DUALITY

Published online by Cambridge University Press:  23 May 2019

Zhenyu Cui ,
Justin Lars Kirkby  and
Duy Nguyen
Zhenyu Cui
Affiliation:
Stevens Institute of Technology
Justin Lars Kirkby*
Affiliation:
Georgia Institute of Technology
Duy Nguyen
Affiliation:
Marist College
*
*Address correspondence to J. Lars Kirkby, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30318, USA; e-mail: jkirkby3@gatech.edu.
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Abstract

In this article, we propose a new nonparametric density estimator derived from the theory of frames and Riesz bases. In particular, we propose the so-called bi-orthogonal density estimator based on the class of B-splines and derive its theoretical properties, including the asymptotically optimal choice of bandwidth. Detailed theoretical analysis and comparisons of our estimator with existing local basis and kernel density estimators are presented. The estimator is particularly well suited for high-frequency data analysis in financial and economic markets.

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ARTICLES
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Econometric Theory , Volume 36 , Issue 2 , April 2020 , pp. 250 - 291
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Copyright © Cambridge University Press 2019 

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References

REFERENCES

Adusumilli, K. & Otsu, T. (2018) Nonparametric instrumental regression with errors in variables. Econometric Theory 36(6), 1256–1280.10.1017/S0266466617000469CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64(3), 527–560.10.2307/2171860CrossRefGoogle Scholar
Bandi, F.M. & Moloche, G. (2017) On the functional estimation of multivariate diffusion processes. Econometric Theory 34(4), 896–946.10.1017/S0266466617000305CrossRefGoogle Scholar
Beare, B. & Schmidt, L. (2014) An empirical test of pricing kernel monotonicity. Journal of Applied Econometrics 31(2), 338–356.10.1002/jae.2422CrossRefGoogle Scholar
Botev, Z.I., Grotowski, J.F., & Kroese, D.P. (2010) Kernel density estimation via diffusion. Annals of Statistics 38(5), 2916–2957.10.1214/10-AOS799CrossRefGoogle Scholar
Bowman, A. (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71, 353–360.10.1093/biomet/71.2.353CrossRefGoogle Scholar
Cai, Z. & Wang, X. (2008) Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics 147(1), 120–130.CrossRefGoogle Scholar
Calonico, S., Cattaneo, M.D., & Farrell, M.H. (2018) On the effect of bias estimation on coverage accuracy in nonparametric inference. Journal of the American Statistical Association 113(522), 767–779.10.1080/01621459.2017.1285776CrossRefGoogle Scholar
Carroll, R., Delaigle, A., & Hall, P. (2013) Unexpected properties of bandwidth choice when smoothing discrete data from construction a functional data classifier. Annals of Statistics 41(6), 2739–2767.10.1214/13-AOS1158CrossRefGoogle Scholar
Cattaneo, M.D., Jansson, M., & Ma, X. (2017) Simple Local Polynomial Density Estimators. Technical report, Working paper. Retrieved July 22, 2017 from http://wwwpersonal.umich.edu/∼cattaneo/papers/Cattaneo-Jansson-Ma_2017_LocPolDensity.pdf.Google Scholar
Cencov, N. (1962) Evaluation of an unknown distribution density from observations. Soviet Mathematics 3, 1559–1562.Google Scholar
Chen, S. & Xu, Z. (2014) On implied volatility for options-some reasons to smile and more to correct. Journal of Econometrics 179(1), 1–15.10.1016/j.jeconom.2013.10.007CrossRefGoogle Scholar
Christensen, O. (2003) An Introduction to Frames and Riesz Bases. Birkhauser Boston.10.1007/978-0-8176-8224-8CrossRefGoogle Scholar
Cui, Z., Kirkby, J., & Nguyen, D. (2017a) Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps. Insurance: Mathematics and Economics 74, 46–62.Google Scholar
Cui, Z., Kirkby, J., & Nguyen, D. (2017b) A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. European Journal of Operational Research 262(1), 381–400.10.1016/j.ejor.2017.04.007CrossRefGoogle Scholar
Donoho, D., Johnstone, I., Kerkyacharian, G., & Picard, D. (1996) Density estimation by wavelet thresholding. Annals of Statistics 24(2), 508–539.Google Scholar
Duin, R. (1976). On the choice of smoothing parameters of Parzen estimators of probability density functions. IEEE Transactions on Computers C-25, 1175–1179.CrossRefGoogle Scholar
Eilers, P. & Marx, B. (1996) Flexible smoothing with B-splines and penalties. Statistical Science 11(2), 89–121.10.1214/ss/1038425655CrossRefGoogle Scholar
Fan, J. (2005) A selective overview of nonparametric methods in financial econometrics. Statistical Science 317–337.10.1214/088342305000000412CrossRefGoogle Scholar
Figueroa-López, J.E. & Li, C. (2016) Optimal kernel estimation of spot volatility of stochastic differential equations, arXiv preprint, arXiv:1612.04507.Google Scholar
Filipović, D., Mayerhofer, E., & Schneider, P. (2013) Density approximations for multivariate affine jump-diffusion processes. Journal of Econometrics 176(2), 93–111.10.1016/j.jeconom.2012.12.003CrossRefGoogle Scholar
Fix, E. & Hodges, J. (1951) Nonparametric discrimination: Consistency properties. Report Number 4, Project Number 21-49-004, USAF School of Aviation Medicine, Randolph Field, Texas, February.Google Scholar
Gehringer, K.R. & Redner, R.A. (1992) Nonparametric probability density estimation using normalized b–splines. Communications in Statistics-Simulation and Computation 21(3), 849–878.10.1080/03610919208813053CrossRefGoogle Scholar
Gu, C. (1993) Smoothing spline density estimation: A dimensionless automatic algorithm. Journal of the American Statistical Association 88(422), 495–504.10.1080/01621459.1993.10476300CrossRefGoogle Scholar
Gu, C. & Qiu, C. (1993) Smoothing spline density estimation: Theory. Annals of Statistics 21(1), 217–234.10.1214/aos/1176349023CrossRefGoogle Scholar
Gu, C. & Wang, J. (2003) Penalized likelihood density estimation: Direct cross-validation and scalable approximation. Statistica Sinica 13, 811–826.Google Scholar
Hall, P. (1981) On trigonometric series estimates of densities. Annals of Statistics 9(3), 683–685.10.1214/aos/1176345474CrossRefGoogle Scholar
Hall, P. (1987) Cross-validation and the smoothing of orthogonal series density estimators. Journal of Multivariate Analysis 21(2), 189–206.10.1016/0047-259X(87)90001-7CrossRefGoogle Scholar
Hall, P., Kang, K.-H. (2005) Bandwidth choice for nonparametric classification. The Annals of Statistics 33(1), 284–30610.1214/009053604000000959CrossRefGoogle Scholar
Hall, P.G. & Racine, J.S. (2015) Infinite order cross-validated local polynomial regression. Journal of Econometrics 185(2), 510–525.CrossRefGoogle Scholar
Hardle, W. & Marron, J.S. (1985) Optimal bandwidth selection in nonparametric regression function estimation. Annals of Statistics 13(4), 1465–1481.10.1214/aos/1176349748CrossRefGoogle Scholar
Hayfield, T., Racine, J.S. (2008) Nonparametric econometrics: The NP package. Journal of Statistical Software 27(5), 1–32.10.18637/jss.v027.i05CrossRefGoogle Scholar
Heil, C. (2011) A Basis Theory Primer, Expanded Edition. Birkhauser.10.1007/978-0-8176-4687-5CrossRefGoogle Scholar
Huang, S.-Y. (1999) Density estimation by wavelet-based reproducing kernels. Statistica Sinica 9(1), 137–151.Google Scholar
Izenman, A. (1991) Recent developments in nonparametric density estimation. Journal of the American Statistical Association 86(413), 205–223.Google Scholar
Jones, M., Marron, J.S., & Park, B.U. (1991) A simple root 𝑛 bandwidth selector. Annals of Statistics 19(4), 1919–1932.10.1214/aos/1176348378CrossRefGoogle Scholar
Jones, M.C., Marron, J.S., & Sheather, S.J. (1996a) A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association 91(433), 401–407.10.1080/01621459.1996.10476701CrossRefGoogle Scholar
Jones, M.C., Marron, J.S., & Sheather, S.J. (1996b) Progress in data-based bandwidth selection for kernel density estimation. Computational Statistics 11(3), 337–381.Google Scholar
Kirkby, J. (2015) Efficient option pricing by frame duality with the fast Fourier transform. SIAM Journal on Financial Mathematics 6(1), 713–747.10.1137/140989480CrossRefGoogle Scholar
Kirkby, J. (2016). An efficient transform method for Asian option pricing. SIAM Journal on Financial Mathematics 7(1), 845–892.10.1137/16M1057127CrossRefGoogle Scholar
Kirkby, J. (2017) Robust option pricing with characteristic functions and the B-spline order of density projection. Journal of Computational Finance 21(2), 101–127.Google Scholar
Kirkby, J. & Deng, S. (2019) Static hedging and pricing of exotic options with payoff frames. Mathematical Finance, 29(2), 407–693.10.1111/mafi.12184CrossRefGoogle Scholar
Kirkby, J., Nguyen, D., & Cui, Z. (2017) A unified approach to Bermudan and barrier options under stochastic volatility models with jumps. Journal of Economic Dynamics and Control 80, 75–100.10.1016/j.jedc.2017.05.001CrossRefGoogle Scholar
Koo, J. (1996) Bivariate B-splines for tensor logspline density estimation. Computational Statistics & Data Analysis 21, 31–42.10.1016/0167-9473(95)00003-8CrossRefGoogle Scholar
Kooperberg, C. & Stone, C. (1991) A study of logspline density estimation. Computational Statistics & Data Analysis 12, 327–347.10.1016/0167-9473(91)90115-ICrossRefGoogle Scholar
Kou, S.G. (2002) A jump-diffusion model for option pricing. Management Science 48(8), 1086–1101.10.1287/mnsc.48.8.1086.166CrossRefGoogle Scholar
Leitao, A., Oosterlee, C., Ortiz-Gracia, L., & Bohte, S. (2018) On the data-driven COS method. Applied Mathematics and Computation 317, 68–84.10.1016/j.amc.2017.09.002CrossRefGoogle Scholar
Li, C. & Chen, D. (2016) Estimating jump–diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1), 51–70.CrossRefGoogle Scholar
Liu, G. & Hong, L.J. (2009) Kernel estimation of quantile sensitivities. Naval Research Logistics 56(6), 511–525.CrossRefGoogle Scholar
Liu, G. & Hong, L.J. (2011) Kernel estimation of the Greeks for options with discontinuous payoffs. Operations Research 59(1), 96–108.10.1287/opre.1100.0844CrossRefGoogle Scholar
Loader, C. (1999) Bandwidth selection: Classical or plug-in? Annals of Statistics 27(2), 415–438.10.1214/aos/1018031201CrossRefGoogle Scholar
Marron, J. (1987) A comparison of cross-validation techniques in density estimation. Annals of Statistics 15(1), 152–162.10.1214/aos/1176350258CrossRefGoogle Scholar
Marron, J. & Wand, M. (1992) Exact mean integrated squared error. Annals of Statistics 20(2), 712–736.10.1214/aos/1176348653CrossRefGoogle Scholar
Marron, J.S. (1985) An asymptotically efficient solution to the bandwidth problem of kernel density estimation. Annals of Statistics 13(3), 1011–1023.10.1214/aos/1176349653CrossRefGoogle Scholar
Masri, R. & Redner, R. (2005) Convergence rates for uniform B-spline density estimators on bounded and semi-infinite domains. Nonparametric Statistics 17(5), 555–582.CrossRefGoogle Scholar
Merton, R. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, 125–144.CrossRefGoogle Scholar
Opschoor, A., Dijk, D., & van der Wel, M. (2017) Combining density forecasts using focused scoring rules. Journal of Applied Econometrics 32(7), 1298–1313.10.1002/jae.2575CrossRefGoogle Scholar
Parzen, E. (1962) On estimation of a probability density function and mode. Annals of Mathematical Statistics 33, 1065–1076.CrossRefGoogle Scholar
Penev, S. & Dechevsky, L. (1997) On nonnegative wavelet-based density estimators. Journal of Nonparametric Statistics 7, 365–394.10.1080/10485259708832711CrossRefGoogle Scholar
Peter, A. & Rangarajan, A. (2008) Maximum likelihood wavelet density estimation with applications to image and shape matching. IEEE Transactions on Image Processing 17(4), 458–468.10.1109/TIP.2008.918038CrossRefGoogle ScholarPubMed
Racine, J. & Li, K. (2017) Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach. Journal of Econometrics 201(1), 72–94.CrossRefGoogle Scholar
Redner, R. (1999) Convergence rates for uniform B-spline density estimators part I: One dimension. SIAM Journal on Scientific Computing 20(6), 1929–1953.10.1137/S1064827595291996CrossRefGoogle Scholar
Rosenblatt, M. (1956) Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics 27, 832–837.10.1214/aoms/1177728190CrossRefGoogle Scholar
Rudemo, M. (1982) Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics 9, 65–78.Google Scholar
Ruijter, M., Versteegh, M., & Oosterlee, C. (2015) On the application of spectral filters in a Fourier option pricing technique. Journal of Computational Finance 19(1), 76–106.10.21314/JCF.2015.306CrossRefGoogle Scholar
Schwartz, S. (1967) Estimation of a probability density by an orthogonal series. Annals of Mathematical Statistics 38, 1261–1265.10.1214/aoms/1177698795CrossRefGoogle Scholar
Sheather, S.J. & Jones, M.C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society. Series B (Methodological) 53(3), 683–690.10.1111/j.2517-6161.1991.tb01857.xCrossRefGoogle Scholar
Silverman, B. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall.CrossRefGoogle Scholar
Song, Z. & Xiu, D. (2016) A tale of two option markets: Pricing kernels and volatility risk. Journal of Econometrics 190(1), 176–196.10.1016/j.jeconom.2015.06.024CrossRefGoogle Scholar
Terrell, G. & Scott, D. (1992) Variable kernel density estimation. Annals of Statistics 20(3), 1236–1265.CrossRefGoogle Scholar
Unser, M. (1996) Vanishing moments and the approximation power of wavelet expansions. In Delogne, P. (ed.), Image Processing, 1996. Proceedings, International Conference on, vol. 1, pp. 629–632. IEEE.Google Scholar
Unser, M. & Daubechies, I. (1997) On the approximation power of convolution-based least squares versus interpolation. IEEE Transactions on Signal Processing 45(7), 1697–1711.CrossRefGoogle Scholar
Van Es, B., Spreij, P., & Zanten, H. (2003) Nonparametric volatility density estimation. Bernoulli 9(3), 451–465.10.3150/bj/1065444813CrossRefGoogle Scholar
Wahba, G. (1981) Data-based optimal smoothing of orthogonal series density estimates. Annals of Statistics 9(1), 146–156.10.1214/aos/1176345341CrossRefGoogle Scholar
Walter, G. & Blum, J. (1979) Probability density estimation using delta sequences. Annals of Statistics 7(2), 328–340.CrossRefGoogle Scholar
Wand, M.P. & Jones, M.C. (1994) Kernel Smoothing. CRC Press.10.1201/b14876CrossRefGoogle Scholar
Wang, C.-S. & Zhao, Z. (2016) Conditional value-at-risk: Semiparametric estimation and inference. Journal of Econometrics 195(1), 86–103.CrossRefGoogle Scholar
Wang, X. & Hong, Y. (2018) Characteristic function based testing for conditional independence: A nonparametric regression approach. Econometric Theory 34(4), 815–849.10.1017/S026646661700010XCrossRefGoogle Scholar
Watson, G. (1969) Density estimation by orthogonal series. The Annals of Mathematical Statistics 38, 1262–1265.Google Scholar
Wegman, E. (1972) Nonparametric probability density estimation: A summary of available methods. Technometrics 14(3), 533–546.10.1080/00401706.1972.10488943CrossRefGoogle Scholar
Woodroofe, M. (1970) On choosing a delta sequence. Annals of Mathematical Statistics 41, 1665–1671.10.1214/aoms/1177696810CrossRefGoogle Scholar
Young, R. (1980) An Introduction to Nonharmonic Fourier Series, Revised ed. Academic Press.Google Scholar
Yu, J. (2007) Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan. Journal of Econometrics 141(2), 1245–1280.10.1016/j.jeconom.2007.02.003CrossRefGoogle Scholar
Zhang, X., Brooks, R., & King, M. (2009) A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation. Journal of Econometrics 153(1), 21–32.10.1016/j.jeconom.2009.04.004CrossRefGoogle Scholar
Zu, Y. (2015) Nonparametric specification tests for stochastic volatility models based on volatility density. Journal of Econometrics 187(1), 323–344.CrossRefGoogle Scholar