Summary
Let h o, ĥ o and ĥ c be the windows which minimise mean integrated square error, integrated square error and the least-squares cross-validatory criterion, respectively, for kernel density estimates. It is argued that ĥ o, not h o, should be the benchmark for comparing different data-driven approaches to the determination of window size. Asymptotic properties of h o-ĥ o and ĥ c -ĥ o, and of differences between integrated square errors evaluated at these windows, are derived. It is shown that in comparison to the benchmark ĥ o, the observable window ĥ c performs as well as the so-called “optimal” but unattainable window h o, to both first and second order.
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References
Bickel, P.J., Rosenblatt, M.: On some global measures of the deviations of density estimates. Ann. Stat. 1, 1071–1095 (1973)
Bowman, A.W.: A comparative study of some kernel-based nonparametric density estimates. Manchester-Sheffield School of Probability and Statistics Research Report 84/AWB/1 (1982)
Bowman, A.W.: An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71, 353–360 (1984)
Bowman, A.W., Hall, P., Titterington, D.M.: Cross-validation in nonparametric estimation of probabilities and probability densities. Biometrika 71, 341–352 (1984)
Chow, Y.S., Geman, S., Wu, L.D.: Consistent cross-validated density estimation. Ann. Stat. 11, 25–38 (1983)
Csörgö, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981
Devroye, L., Györfi, L.: Nonparametric density estimation: the L 1 view. New York: Wiley 1985
Duin, R.P.W.: On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Trans. Comput. C25, 1175–1179 (1976)
Habbema, J.D.F., Hermans, J., van den Broek, K.: A stepwise discriminant analysis program using density estimation. Compstat 1974, ed.: G. Bruckman 101–110. Vienna: Physica 1974
Hall, P.: Limit theorems for stochastic measures of the accuracy of density estimators. Stochastic Processes Appl. 13, 11–25 (1982)
Hall, P.: Large sample optimality of least squares cross-validation in density estimation. Ann. Stat. 11, 1156–1174 (1983)
Hall, P.: Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal. 14, 1–16 (1984)
Hall, P.: Asymptotic theory of minimum integrated square error for multivariate density estimation. In: Krishnaiah, P.R. (ed.) Proc. Sixth Internat. Sympos. Multivariate Analysis, pp 289–309. Amsterdam: North Holland 1985
Marron, J.S.: An asymptotically efficient solution to the bandwidth problem of kernel density estimation. Ann. Stat. 13, 1011–1023 (1985).
Marron, J.S.: A comparison of cross-validation techniques in density estimation. N.C. Inst. of Statist. Mimeo Series #1568 (1985)
Rice, J.: Bandwidth choice for nonparametric regression. Ann. Stat. 12, 1215–1230 (1984)
Rosenblatt, M.: Curve estimates. Ann. Math. Stat. 42, 1815–1842 (1971)
Rosenblatt, M.: A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann. Stat. 3, 1–14 (1975)
Rudemo, M.: Empirical choice of histogram and kernel density estimators. Scand. J. Stat. 9, 65–78 (1982)
Silverman, B.W.: Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Stat. 6, 177–184 (1978)
Stone, C.J.: An asymptotically optimal window selection rule for kernel density estimates. Ann. Stat. 12, 1285–1297 (1984)
Titterington, D.M.: Common structure of smoothing techniques in statistics. Int. Stat. Rev. 53, 141–170 (1985)
Woodroofe, M.: On choosing a delta sequence. Ann. Math. Stat. 41, 1665–1671 (1970)
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On leave from Australian National University. — Work of first author supported by U.S.A.F. Grant No. F 49620 82 C 009.
Research of second author partially supported by NSF Grant DMS-8400602.
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Hall, P., Marron, J.S. Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation. Probab. Th. Rel. Fields 74, 567–581 (1987). https://doi.org/10.1007/BF00363516
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DOI: https://doi.org/10.1007/BF00363516