Nonparametric density estimation and bandwidth selection with B-spline bases: A novel Galerkin method

Ahlberg, 1963, Convergence properties of the spline fit, J. Soc. Ind. Appl. Math., 11, 95, 10.1137/0111007 Aït-Sahalia, 2016, Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models, J. Econometrics, 192, 119, 10.1016/j.jeconom.2015.11.002 Bhattacharya, 2010, Nonparametric Bayesian density estimation on manifolds with applications to planar shapes, Biometrika, 97, 851, 10.1093/biomet/asq044 Botev, 2010, Kernel density estimation via diffusion, Ann. Statist., 38, 2916, 10.1214/10-AOS799 Bowman, 1984, An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71, 353, 10.1093/biomet/71.2.353 Bowman, 1984, Cross-validation in nonparametric estimation of probabilities and probability densities, Biometrika, 71, 341, 10.1093/biomet/71.2.341 Brunk, 1978, Univariate density estimation by orthogonal series, Biometrika, 65(3), 521, 10.1093/biomet/65.3.521 Carroll, 2013, Unexpected properties of bandwidth choice when smoothing discrete data from construction a functional data classifier, Ann. Statist., 41(6), 2739 Céa, 1964, Approximation variationnelle des problèmes aux limites, 14, 345 Cencov, 1962, Evaluation of an unknown distribution density from observations, Soviet Math., 3, 1559 Cheng, 2008, Kernel methods for optimal change-points estimation in derivatives, J. Comput. Graph. Statist., 17, 56, 10.1198/106186008X289164 Christensen, 2003 Ciarlet, 2002 Colombo, 2018, Uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins, Comput. Methods Appl. Mech. Engrg., 328, 122, 10.1016/j.cma.2017.08.049 Cui, 2020, Nonparametric density estimation by B-spline duality, Econometric Theory, 1 Cui, 2021, A data-driven framework for consistent financial valuation and risk measurement, European J. Oper. Res., 289(1), 381, 10.1016/j.ejor.2020.07.011 Cui, 2021, Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations, European J. Oper. Res., 290(3), 1046, 10.1016/j.ejor.2020.09.008 Dai, 2017, Optimal Bayes classifiers for functional data and density ratios, Biometrika, 104, 545 Ditkowski, 2020, Density estimation in uncertainty propagation problems using a surrogate model, SIAM/ASA J. Uncertain. Quantif., 8, 261, 10.1137/18M1205959 Donoho, 1996, Density estimation by wavelet thresholding, Ann. Statist., 24(2), 508 Durrett, 2010 Eilers, 1996, Flexible smoothing with B-splines and penalties, Statist. Sci., 11(2), 89 Fan, 1996 Fix, 1951 Gu, 1993, Smoothing spline density estimation: a dimensionless automatic algorithm, J. Amer. Statist. Assoc., 88(422), 495, 10.1080/01621459.1993.10476300 Gu, 1993, Smoothing spline density estimation: theory, Ann. Statist., 21(1), 217 Hall, 1981, On trigonometric series estimates of densities, Ann. Statist., 9, 683, 10.1214/aos/1176345474 Hall, 1982, Cross-validation in density estimation, Biometrika, 69, 383, 10.1093/biomet/69.2.383 Hall, 1987, Cross-validation and the smoothing of orthogonal series density estimators, J. Multivariate Anal., 21, 189, 10.1016/0047-259X(87)90001-7 Hall, 2005, Bandwidth choice for nonparametric classification, Ann. Statist., 33, 284, 10.1214/009053604000000959 Hall, 1991, On optimal data-based bandwidth selection in kernel density estimation, Biometrika, 78, 263, 10.1093/biomet/78.2.263 Heil, 2011 Herrmann, 1997, Local bandwidth choice in kernel regression estimation, J. Comput. Graph. Statist., 6, 35 Horn, 2012 Huang, 1999, Density estimation by wavelet-based reproducing kernels, Statist. Sinica, 9, 137 Izenman, 1991, Recent developments in nonparametric density estimation, J. Amer. Statist. Assoc., 86(413), 205 Jones, 1996, A brief survey of bandwidth selection for density estimation, J. Amer. Statist. Assoc., 91, 401, 10.1080/01621459.1996.10476701 Jones, 1996, Progress in data-based bandwidth selection for kernel density estimation, Comput. Statist., 11, 337 Kirkby, 2015, Efficient option pricing by frame duality with the fast fourier transform, SIAM J. Financial Math., 6(1), 713, 10.1137/140989480 Kirkby, 2017, Robust option pricing with characteristic functions and the B-spline order of density projection, J. Comput. Finance, 21(2), 101 Kirkby, 2019, Static hedging and pricing of exotic options with payoff frames, Math. Finance, 29(2), 612, 10.1111/mafi.12184 Kirkby, 2020, An analysis of dollar cost averaging and market timing investment strategies, European J. Oper. Res., 286(3), 1168, 10.1016/j.ejor.2020.04.055 Koo, 1996, Bivariate B-splines for tensor logspline density estimation, Comput. Statist. Data Anal., 21, 31, 10.1016/0167-9473(95)00003-8 Kooperberg, 1991, A study of logspline density estimation, Comput. Statist. Data Anal., 12, 327, 10.1016/0167-9473(91)90115-I Kooperberg, 1992, Logspline density estimation for censored data, J. Comput. Graph. Statist., 1, 301 Kooperberg, 2004, Comparison of parametric and bootstrap approaches to obtaining confidence intervals for logspline density estimation, J. Comput. Graph. Statist., 1, 106, 10.1198/1061860043047 Lai, 2007 Leitao, 2018, On the data-driven COS method, Appl. Math. Comput., 317, 68, 10.1016/j.amc.2017.09.002 Leitao, 2020, Model-free computation of risk contributions in credit portfolios, Appl. Math. Comput., 382, 10.1016/j.amc.2020.125351 Loader, 1999, Bandwidth selection: classical or plug-in?, Ann. Statist., 27(2), 415 Marron, 1985, An asymptotically efficient solution to the bandwidth problem of kernel density estimation, Ann. Statist., 13, 1011, 10.1214/aos/1176349653 Masdemont, 2014, Haar wavelets-based approach for quantifying credit portfolio losses, Quant. Finance, 14, 1587, 10.1080/14697688.2011.595731 Matthies, 2005, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 194, 1295, 10.1016/j.cma.2004.05.027 McCloud, 2020, Determining the number of effective parameters in kernel density estimation, Comput. Statist. Data Anal., 143, 10.1016/j.csda.2019.106843 Morača, 2008, Bounds for norms of the matrix inverse and the smallest singular value, Linear Algebra Appl., 429, 2589, 10.1016/j.laa.2007.12.026 Muller, 1998, Bayesian inference with wavelets: Density estimation, J. Comput. Graph. Statist., 7, 456 Ortiz-Gracia, 2014, Efficient VaR and expected shortfall computations for nonlinear portfolios within the delta-gamma approach, Appl. Math. Comput., 244, 16, 10.1016/j.amc.2014.06.110 Papp, 2014, Shape-constrained estimation using nonnegative splines, J. Comput. Graph. Statist., 23, 211, 10.1080/10618600.2012.707343 Parzen, 1962, On estimation of a probability density function and mode, Ann. Math. Stat., 33, 1065, 10.1214/aoms/1177704472 Penev, 1997, On non-negative wavelet-based density estimators, J. Nonparametr. Stat., 7, 365, 10.1080/10485259708832711 Peter, 2008, Maximum likelihood wavelet density estimation with applications to image and shape matching, IEEE Trans. Image Process., 17(4), 458, 10.1109/TIP.2008.918038 Racine, 2017, Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach, J. Econometrics, 201, 72, 10.1016/j.jeconom.2017.06.020 Rahman, 2020, A spline chaos expansion, SIAM/ASA J. Uncertain. Quantif., 8, 27, 10.1137/19M1239702 Rathke, 2019, Fast multivariate log-concave density estimation, Comput. Statist. Data Anal., 140, 41, 10.1016/j.csda.2019.04.005 Redner, 1999, Convergence rates for uniform B-spline density estimators part I: one dimension, SIAM J. Sci. Comput., 20(6), 1929, 10.1137/S1064827595291996 Rosenblatt, 1956, Remarks on some nonparametric estimates of a density function, Ann. Math. Stat., 27, 832, 10.1214/aoms/1177728190 Rudemo, 1982, Empirical choice of histograms and kernel density estimators, Scand. J. Stat., 9, 65 Schwartz, 1967, Estimation of a probability density by an orthogonal series, Ann. Math. Stat., 38, 1261, 10.1214/aoms/1177698795 Scott, 1987, Biased and unbiased cross-validation in density estimation, J. Amer. Stat. Assoc., 82, 1131, 10.1080/01621459.1987.10478550 Sheather, 2004, Density estimation, Statist. Sci., 19, 588, 10.1214/088342304000000297 Sheather, 1991, A reliable data-based bandwidth selection method for kernel density estimation, J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 683 Treviño, 2019, The radial wavelet frame density estimator, Comput. Statist. Data Anal., 130, 111, 10.1016/j.csda.2018.08.021 Tsybakov, 2008 Unser, 1996, Vanishing moments and the approximation power of wavelet expansions, 629 Unser, 1997, On the approximation power of convolution-based least squares versus interpolation, IEEE Trans. Signal Process., 45, 1697, 10.1109/78.599940 Wahba, 1981, Data-based optimal smoothing of orthogonal series density estimates, Ann. Statist., 9, 146, 10.1214/aos/1176345341 Wand, 1994, Fast computation of multivariate kernel estimators, J. Comput. Graph. Statist., 3, 433 Wand, 1994 Wang, 2019, Computing the Gerber–Shiu function by frame duality projection, Scand. Actuar. J., 4, 291, 10.1080/03461238.2018.1557739 Watson, 1969, Density estimation by orthogonal series, Ann. Math. Stat., 38, 1262 Wegman, 1972, Nonparametric probability density estimation: A summary of available methods, Technometrics, 14(3), 533, 10.1080/00401706.1972.10488943 Xie, 2020 Young, 1980 Zhang, 2020, Valuing equity-linked death benefits in general exponential Lévy models, J. Comput. Appl. Math., 365, 10.1016/j.cam.2019.112377