×

zbMATH — the first resource for mathematics

A review and some new proposals for bandwidth selection in nonparametric density estimation for dependent data. (English) Zbl 1383.62095
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 173-208 (2017).
Summary: When assuming independence, the choice of the smoothing parameter in density estimation has been extensively studied. However, when considering dependence, very few papers have dealt with data-driven bandwidth selectors. First of all, we review the state of art of the existing methods for bandwidth selection under dependence. Moreover, three new (or adapted) methods are proposed: (a) an extension to the dependent case of the modified cross-validation by W. Stute [J. Stat. Plann. Inference 30, No. 3, 293–305 (1992; Zbl 0756.62019)]; (b) an adaptation to density estimation of the penalized cross-validation proposed by G. Estévez-Pérez et al. [ibid. 104, No. 1, 1–30 (2002; Zbl 0988.62021)] for hazard rate estimation; (c) finally, the smoothed version of the so-called moving blocks bootstrap is established and an exact expression for the bootstrap version of the mean integrated squared error under dependence is obtained in this context. This is useful since Monte Carlo approximation is not needed to implement the bootstrap selector. An extensive simulation study is carried out in order to check and compare the empirical behaviour of six selected bandwidths.
For the entire collection see [Zbl 1383.62010].

MSC:
62G07 Density estimation
62G09 Nonparametric statistical resampling methods
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barbeito I, Cao R (2016) Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data. Comput Statist & Data Anal 104:130-147 · Zbl 06918032
[2] Bowman A (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71:353-360
[3] Cao R (1993) Bootstrapping the mean integrated squared error. J Mult Anal 45:137-160 · Zbl 0779.62038
[4] Cao R (1999) An overview of bootstrap methods for estimating and predicting in time series. Test 8:95-116 · Zbl 0945.62048
[5] Cao R, Quintela del Río A, Vilar Fernández J (1993) Bandwidth selection in nonparametric density estimation under dependence. a simulation study. Com Statist 8:313-332 · Zbl 0942.62037
[6] Cao R, Cuevas A, González Manteiga W (1994) A comparative-study of several smoothing methods in density-estimation. Comput Statist & Data Anal 17:153-176 · Zbl 0937.62518
[7] Chow Y, Geman S, Wu L (1983) Consistent cross-validated density-estimation. Ann Statist 11:25-38 · Zbl 0509.62033
[8] Cox D, Kim T (1997) A study on bandwidth selection in density estimation under dependence. J Mult Anal 62:190-203 · Zbl 0896.62033
[9] Devroye L (1987) A Course in Density Estimation. Birkhauser, Boston · Zbl 0617.62043
[10] Efron B (1979) Bootstrap methods: Another look at the jackniffe. Ann Statist 7:1-26 · Zbl 0406.62024
[11] Efron B, Tibishirani R (1993) An Introduction to the Bootstrap. Chapman and Hall, New York
[12] Estévez-Pérez G, Quintela del Río A, Vieu P (2002) Convergence rate for cross-validatory bandwidth in kernel hazard estimation from dependent samples. J Statisti Plann Infer 104:1-30 · Zbl 0988.62021
[13] Faraway J, Jhun M (1990) Bootstrap choice of bandwidth for density estimation. J Amer Statist Assoc 85:1119-1122
[14] Feluch W, Koronacki J (1992) A note on modified cross-validation in density-estimation. Comput Statist & Data Anal 13:143-151 · Zbl 0850.62343
[15] Hall P (1983) Large sample optimality of least-squares cross-validation in densityestimation. Ann Statist 11:1156-1174 · Zbl 0599.62051
[16] Hall P (1990) Using the bootstrap to estimate mean squared error and select smoothing parameter in nonprametric problems. J Multiv Anal 32:177-203 · Zbl 0722.62030
[17] Hall P, Marron J (1987a) Extent to which least-squares cross-validation minimizes integrated square error in nonparametric density-estimation. Probab Theor Relat Fields 74:567-581 · Zbl 0588.62052
[18] Hall P, Marron J (1987b) On the amount of noise inherent in bandwidth selection for a kernel density estimator. Ann Statist 15:163-181 · Zbl 0667.62022
[19] Hall P, Marron J (1991) Local minima in cross-validation functions. J Roy Statist Soc Ser B 53:245-252 · Zbl 0800.62216
[20] Hall P, Lahiri S, Truong Y (1995) On bandwidth choice for density estimation with dependent data. Ann Statist 23:2241-2263 · Zbl 0854.62039
[21] Hart J, Vieu P (1990) Data-driven bandwidth choice for density estimation based on dependent data. Ann Statist 18:873-890 · Zbl 0703.62045
[22] Jones M, Marron J, Park B (1991) A simple root-n bandwidth selector. Ann Statist 19:1919-1932 · Zbl 0745.62033
[23] Jones M, Marron J, Sheather S (1996) A brief survey of bandwidth selection for density estimation. J Amer Statist Assoc 91:401-407 · Zbl 0873.62040
[24] Kreiss J, Paparoditis E (2011) Bootstrap methods for dependent data: A review. J Korean Statist Soc 40:357-378 · Zbl 1296.62172
[25] Künsch H (1989) The jackknife and the bootstrap for general stationary observations. Ann Statist 17:1217-1241 · Zbl 0684.62035
[26] Léger C, Romano J (1990) Bootstrap choice of tuning parameters. Ann Inst Statist Math 42:709-735 · Zbl 0722.62032
[27] Liu R, Singh K (1992) Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap, eds R LePage and L Billard pp 225-248 · Zbl 0838.62036
[28] Marron J (1985) An asymptotically efficient solution to the bandwidth problem of kernel density-estimation. Ann Statist 13:1011-1023 · Zbl 0585.62073
[29] Marron J (1987) A comparison of cross-validation techniques in density estimation. Ann Statist 15:152-162 A review and new proposals for bandwidth selection in density estimation under dependence 31 · Zbl 0619.62032
[30] Marron J (1992) Bootstrap bandwidth selection. In Exploring the Limits of Bootstrap, eds R LePage and L Billard pp 249-262 · Zbl 0838.62029
[31] Marron J, Wand M (1992) Exact mean integrated squared error. Ann Statist 20(2):712-736 · Zbl 0746.62040
[32] Park B, Marron J (1990) Comparison of data-driven bandwidth selectors. J Amer Statist Assoc 85:66-72
[33] Parzen E (1962) Estimation of a probability density-function and mode. Ann Math Statist 33:1065-1076 · Zbl 0116.11302
[34] Politis D, Romano J (1994) The stationary bootstrap. J Amer Statist Assoc 89:1303-1313 · Zbl 0814.62023
[35] Rosenblatt M (1956) Estimation of a probability density-function and mode. Ann Math Statist 27:832-837 · Zbl 0073.14602
[36] Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Statist 9:65-78 · Zbl 0501.62028
[37] Scott D, Terrell G (1987) Biased and unbiased cross-validation in density-estimation. J Amer Statist Assoc 82:1131-1146 · Zbl 0648.62037
[38] Sheather S, Jones M (1991) A reliable data-based bandwidth selection method for kernel density estimation. J Roy Statist Soc Ser B 53:683-690 · Zbl 0800.62219
[39] Silverman B, Young G (1987) The bootstrap: To smooth or not to smooth? Biometrika 74:469-479 · Zbl 0654.62034
[40] Silverman BW (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall, London
[41] Stone C (1984) An asymptotically optimal window selection rule for kernel density estimates. Ann Statist 12:1285-1297 · Zbl 0599.62052
[42] Stute W (1992) Modified cross-validation in density-estimation. J Statisti Plann Infer 30:293-305 · Zbl 0756.62019
[43] Taylor C (1989) Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika 76:705-712 · Zbl 0678.62042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.