How to cite this WIREs title:
WIREs Comp Stat

Nonparametric curve estimation and bootstrap bandwidth selection

Advanced Review

Ricardo Cao, Inés Barbeito

Published Online: Oct 27 2019

DOI: 10.1002/wics.1488

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Abstract Over the last four decades, the bootstrap method has been considered so as to define data‐driven bandwidth selectors for nonparametric curve estimation. An extensive and updated review of bootstrap methods used to select the smoothing parameter for the nonparametric estimation of several curves has been carried out. Different data generating processes have been profoundly reviewed, such as the classical independent and identically distributed setup as well as dependent, censored, length‐biased, grouped, missing or directional data, among others. Several curves have also been considered, such as the density, regression, hazard rate, intensity, latency, incidence, or distribution functions, among many others. It is worth mentioning that there exist situations where Monte Carlo methods are not needed when using the bootstrap for bandwidth selections. This idea has also been exploited to find closed expressions for the bootstrap version of criterion functions for some proxy of the real curve estimator. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bootstrap and Resampling Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Statistical and Graphical Methods of Data Analysis > Density Estimation

Nonparametric density estimation for the geyser data set using the smoothing parameters h_SSB (black line), h_SMBB (black line), h_{CV l} (green line) and h_PI (blue line)

[ Normal View | Magnified View ]

References

Arteche,, J., & Orbe,, J. (2009). Bootstrap‐based bandwidth choice for log‐periodogram regression. Journal of Time Series Analysis, 30(6), 591–617. https://doi.org/10.1111/j.1467-9892.2009.00629.x
Azzalini,, A. (1981). A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika, 68(1), 326–328. https://doi.org/10.1093/biomet/68.1.326
Barbeito,, I., & Cao,, R. (2016). Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data. Computational Statistics %26 Data Analysis, 104, 130–147.
Barbeito,, I., & Cao,, R. (2017). A review and some new proposals for bandwidth selection in nonparametric density estimation for dependent data. In D. Ferger,, W. González Manteiga,, T. Schmidt,, & J.‐L. Wang, (Eds.), From statistics to mathematical finance: Festschrift in honour of winfried stute (pp. 173–208). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-50986-0_10
Barbeito,, I., & Cao,, R. (2019). Smoothed bootstrap bandwidth selection for nonparametric hazard rate estimation. Journal of Statistical Computation and Simulation, 89(1), 15–37. https://doi.org/10.1080/00949655.2018.1532512
Barbeito,, I., Cao,, R., & Sperlich,, S. (2019). Bandwidth selection for nonparmetric kernel prediction. Preprint.
Borrajo,, M. I., González‐Manteiga,, W., & Martínez‐Miranda,, M. D. (2017). Bandwidth selection for kernel density estimation with length‐biased data. Journal of Nonparametric Statistics, 29(3), 636–668. https://doi.org/10.1080/10485252.2017.1339309
Bose,, A., & Dutta,, S. (2013). Density estimation using bootstrap bandwidth selector. Statistics %26 Probability Letters, 83(1), 245–256. https://doi.org/10.1016/j.spl.2012.08.027
Bowman,, A. W. (1984). An alternative method of cross‐validation for the smoothing of density estimates. Biometrika, 71, 353–360.
Cao,, R. (1993). Bootstrapping the mean integrated squared error. Journal of Multivariate Analysis, 45, 137–160.
Cao,, R., Cuevas,, A., & González Manteiga,, W. (1994). A comparative‐study of several smoothing methods in density‐estimation. Computational Statistics %26 Data Analysis, 17, 153–176.
Cao,, R., Francisco‐Fernández,, M., Anand,, A., Bastida,, F., & González‐Andújar,, J. L. (2013). Modeling Bromus diandrus seedling emergence using nonparametric estimation. Journal of Agricultural, Biological, and Environmental Statistics, 18(1), 64–86. https://doi.org/10.1007/s13253-012-0122-x
Cao,, R., Quintela del Río,, A., & Vilar Fernández,, J. (1993). Bandwidth selection in nonparametric density estimation under dependence. A simulation study. Computational Statistics, 8, 313–332.
Chacón,, J. E., Montanero,, J., & Nogales,, A. G. (2008). Bootstrap bandwidth selection using an h‐dependent pilot bandwidth. Scandinavian Journal of Statistics, 35(1), 139–157.
Davison,, A. C., & Hinkley,, D. V. (1997). Bootstrap methods and their application. New York, NY: Cambridge University Press.
de Uña‐Álvarez,, J., González‐Manteiga,, W., & Cadarso‐Suárez,, C. (1997). Bootstrap selection of the smoothing parameter in density estimation under the koziol‐green model. Lecture Notes‐Monograph Series, 31, 385–398. http://www.jstor.org/stable/4355993
Delaigle,, A., & Gijbels,, I. (2004a). Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Annals of the Institute of Statistical Mathematics, 56(1), 19–47. https://doi.org/10.1007/BF02530523
Delaigle,, A., & Gijbels,, I. (2004b). Practical bandwidth selection in deconvolution kernel density estimation. Computational Statistics %26 Data Analysis, 45(2), 249–267. https://doi.org/10.1016/S0167-9473(02)00329-8
Devroye,, L. (1987). A course in density estimation. Boston, MA: Birkhauser.
Efron,, B. (1979). Bootstrap methods: Another look at the jackniffe. Annals of Statistics, 7, 1–26.
Efron,, B., & Tibishirani,, R. (1993). An introduction to the bootstrap. New York, NY: Chapman and Hall.
Falk,, M. (1992). Bootstrap optimal bandwidth selection for kernel density estimates. Journal of Statistical Planning and Inference, 30(1), 13–22. https://doi.org/10.1016/0378-3758(92)90103-Y
Faraway,, J. J., & Jhun,, M. (1990). Bootstrap choice of bandwidth for density estimation. Journal of the American Statistical Association, 85, 1119–1122.
Feng,, Y., & Heiler,, S. (2009). A simple bootstrap bandwidth selector for local polynomial fitting. Journal of Statistical Computation and Simulation, 79(12), 1425–1439. https://doi.org/10.1080/00949650802352019
Fuentes‐Santos,, I., González‐Manteiga,, W., & Mateu,, J. (2016). Consistent smooth bootstrap kernel intensity estimation for inhomogeneous spatial poisson point processes. Scandinavian Journal of Statistics, 43(2), 416–435. https://doi.org/10.1111/sjos.12183
García‐Portugués,, E., Crujeiras,, R. M., & González‐Manteiga,, W. (2013). Kernel density estimation for directional–linear data. Journal of Multivariate Analysis, 121, 152–175. https://doi.org/10.1016/j.jmva.2013.06.009
González‐Manteiga,, W., Cao,, R., & Marron,, J. S. (1996). Bootstrap selection of the smoothing parameter in nonparametric hazard rate estimation. Journal of the American Statistical Association, 91(435), 1130–1140.
González‐Manteiga,, W., Martínez‐Miranda,, M. D., & Pérez‐González,, A. (2004). The choice of smoothing parameter in nonparametric regression through wild bootstrap. Computational Statistics %26 Data Analysis, 47(3), 487–515. https://doi.org/10.1016/j.csda.2003.12.007
Grund,, B., & Polzehl,, J. (1997). Bias corrected bootstrap bandwidth selection. Journal of Nonparametric Statistics, 8(2), 97–126. https://doi.org/10.1080/10485259708832716
Hall,, P. (1990). Using the bootstrap to estimate mean squared error and select smoothing parameter in nonprametric problems. Journal of Multivariate Analysis, 32, 177–203.
Hall,, P., Lahiri,, S. N., & Truong,, Y. K. (1995). On bandwidth choice for density estimation with dependent data. Ann. Statist., 23, 2241–2263.
Hall,, P., Marron,, J. S., & Park,, B. U. (1992). Smoothed cross‐validation. Probability Theory and Related Fields, 92(1), 1–20. https://doi.org/10.1007/BF01205233
Hall,, P., Wolff,, R. C. L., & Yao,, Q. (1999). Methods for estimating a conditional distribution function. Journal of the American Statistical Association, 94(445), 154–163. https://doi.org/10.1080/01621459.1999.10473832
Hart,, J. D., & Vieu,, P. (1990). Data‐driven bandwidth choice for density estimation based on dependent data. Ann. Statist., 18, 873–890.
Heidenreich,, N.‐B., Schindler,, A., & Sperlich,, S. (2013). Bandwidth selection for kernel density estimation: A review of fully automatic selectors. Advances in Statistical Analysis, 97(4), 403–433. https://doi.org/10.1007/s10182-013-0216-y
Jácome,, M. A., & Cao,, R. (2008). Asymptotic‐based bandwidth selection for the presmoothed density estimator with censored data. Journal of Nonparametric Statistics, 20(6), 483–506. https://doi.org/10.1080/10485250802280226
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, 401–407.
Jones,, M. C., Marron,, J. S., & Sheather,, S. J. (1996b). Progress in data‐based bandwidth selection for kernel density estimation. Computational Statistics, 11, 337–381.
Köhler,, M., Schindler,, A., & Sperlich,, S. (2014). A review and comparison of bandwidth selection methods for kernel regression. International Statistical Review, 82(2), 243–274. https://doi.org/10.1111/insr.12039
Künsch,, H. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist., 17, 1217–1241.
Liu,, R. Y., & Singh,, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In R. LePage, & L. Billard, (Eds.), Exploring the limits of bootstrap (pp. 225–248). New York, NY: John Wiley.
Loh,, J. M., & Jang,, W. (2010). Estimating a cosmological mass bias parameter with bootstrap bandwidth selection. Journal of the Royal Statistical Society Series C (Applied Statistics), 59(5), 761–779.
López‐Cheda,, A., Cao,, R., & Jácome,, M. A. (2017). Nonparametric latency estimation for mixture cure models. Test, 26(2), 353–376. https://doi.org/10.1007/s11749-016-0515-1
López‐Cheda,, A., Cao,, R., Jácome,, M. A., & Van Keilegom,, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics %26 Data Analysis, 105, 144–165. https://doi.org/10.1016/j.csda.2016.08.002
Mammen,, E., Martínez‐Miranda,, M. D., Nielsen,, J. P., & Sperlich,, S. (2011). Do‐validation for kernel density estimation. Journal of the American Statistical Association, 106, 651–660.
Marron,, J. S. (1992). Bootstrap bandwidth selection. In R. LePage, & L. Billard, (Eds.), Exploring the limits of bootstrap (pp. 249–262). New York, NY: John Wiley.
Marron,, J. S., & Wand,, M. P. (1992). Exact mean integrated squared error. The Annals of Statistics, 20(2), 712–736.
Martínez‐Miranda,, M. D., Raya‐Miranda,, R., González‐Manteiga,, W., & González‐Carmona,, A. (2008). A bootstrap local bandwidth selector for additive models. Journal of Computational and Graphical Statistics, 17(1), 38–55. https://doi.org/10.1198/106186008X284097
Miecznikowski,, J. C., Wang,, D., & Hutson,, A. (2010). Bootstrap MISE estimators to obtain bandwidth for kernel density estimation. Communications in Statistics – Simulation and Computation, 39(7), 1455–1469. https://doi.org/10.1080/03610918.2010.500108
Nadaraya,, E. (1964). On estimating regression. Theory of Probability %26 Its Applications, 9(1), 141–142. https://doi.org/10.1137/1109020
Nishiyama,, Y. (2005). Kernel order selection by minimum bootstrapped MSE for density weighted averages. Mathematics and Computers in Simulation (MATCOM; vol. 69, pp. 113–122). Elsevier.
Park,, B., & Marron,, J. S. (1990). Comparison of data‐driven bandwidth selectors. Journal of the American Statistical Association, 85, 66–72.
Parzen,, E. (1962). Estimation of a probability density‐function and mode. The Annals of Mathematical Statistics, 33, 1065–1076.
Politis,, D., & Romano,, J. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89, 1303–1313.
Raya‐Miranda,, R., & Martínez‐Miranda,, M. D. (2011). Data‐driven local bandwidth selection for additive models with missing data. Applied Mathematics and Computation, 217(24), 10328–10342. https://doi.org/10.1016/j.amc.2011.05.040
Reyes,, M., Francisco‐Fernández,, M., & Cao,, R. (2017). Bandwidth selection in kernel density estimation for interval‐grouped data. Test, 26(3), 527–545. https://doi.org/10.1007/s11749-017-0523-9
Reyes,, M., Francisco‐Fernández,, M., Cao,, R., & Barreiro‐Ures,, D. (2019). Kernel distribution estimator for grouped data. To appear in SORT, 43(2), 2019.
Rosenblatt,, M. (1956). Estimation of a probability density‐function and mode. The Annals of Mathematical Statistics, 27, 832–837.
Rudemo,, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65–78.
Saavedra,, A., & Cao,, R. (2001). Smoothed bootstrap bandwidth selection in nonparametric density estimation for moving average processes. Stochastic Analysis and Applications, 19, 555–580.
Sánchez‐Sellero,, C., González‐Manteiga,, W., & Cao,, R. (1999). Bandwidth selection in density estimation with truncated and censored data. Annals of the Institute of Statistical Mathematics, 51(1), 51–70. https://doi.org/10.1023/A:1003879001416
Scott,, D., & Terrell,, G. (1987). Biased and unbiased cross‐validation in density‐estimation. Journal of the American Statistical Association, 82, 1131–1146.
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, 683–690.
Silverman,, B. W. (1986). Density estimation for statistics and data analysis. London, England: Chapman %26 Hall.
Silverman,, B. W., & Young,, G. A. (1987). The bootstrap: To smooth or not to smooth? Biometrika, 74, 469–479.
Stute,, W. (1992). Modified cross‐validation in density‐estimation. Journal of Statistical Planning and Inference, 30, 293–305.
Subramanian,, S., & Bean,, D. (2008). The missing censoring indicator model and the smoothed bootstrap. Computational Statistics and Data Analysis, 53(2), 471–476. https://doi.org/10.1016/j.csda.2008.08.019
Taylor,, C. C. (1989). Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika, 76, 705–712.
Watson,, G. (1964). Smooth regression analysis. Sankhyā: The Indian Journal of Statistics, Series A (1961–2002), 26(4), 359–372.
Watson,, G. S., & Leadbetter,, M. R. (1964a). Hazard analysis i. Biometrika, 51, 175–184.
Watson,, G. S., & Leadbetter,, M. R. (1964b). Hazard analysis ii. Sankhyā Series A, 26, 101–116.
Żychaluk,, K. (2014). Bootstrap bandwidth selection method for local linear estimator in exponential family models. Journal of Nonparametric Statistics, 26(2), 305–319. https://doi.org/10.1080/10485252.2014.885023

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Nonparametric curve estimation and bootstrap bandwidth selection

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox