Abstract
The differentiability properties of statistical functionals have several interesting applications. We are concerned with two of them. First, we prove a result on asymptotic validity for the so-called smoothed bootstrap (where the artificial samples are drawn from a density estimator instead of being resampled from the original data). Our result can be considered as a smoothed analog of that obtained by Parr (1985, Stat. Probab. Lett., 3, 97-100) for the standard, unsmoothed bootstrap. Second, we establish a result on asymptotic normality for estimators of type \(T_n = T(\hat f_n )\) generated by a density functional \(T = T(f),{\text{ }}\hat f_n \) being a density estimator. As an application, a quick and easy proof of the asymptotic normality of \(\int {\hat f_n^2 } \), (the plug-in estimator of the integrated squared density \(\int {f^2 } \)) is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arcones, M. A. and Giné, E. (1992). On the bootstrap of M-estimators and other statistical functionals, Proc. IMS Bootstrap Conference, East Lansing, 1990.
Aubuchon, C. and Hettmansperger, T. P. (1984). A note on the estimation of the integral of f 2(x), J. Statist. Plann. Inference, 9, 321–331.
Bickel, P. J. and Frcedman, D. A. (1981). Some asymptotic theory for the bootstrap, Ann. Statist., 9, 1196–1217.
Billingsley, P. (1986). Probability and Measure, second edition, Wiley, New York.
Birgé, L. and Massart, P. (1995). Estimation of integral functionals of a density, Ann. Statist., 23, 11–29.
Boos, D. D. and Serfling, R. J. (1980). A note on differentials and the CLT and LIL for statistical functions with application to M-estimates, Ann. Statist., 8, 618–624.
Cao, R. (1990). Aplicaciones y nuevos resultados del método bootstrap en la estimación no paramétrica de curvas, Ph.D. Thesis, Universidad de Santiago de Compostela.
Cao, R. (1993). Bootstrapping the mean integrated squared error, J. Multivariate Anal., 45, 137–160.
Clarke, B. R. (1986). Nonsmooth analysis and Fréchet differentiability of M-functionals, Probab. Theory Related Fields, 73, 197–209.
Datta, S. (1992). Some non-asymptotic bounds for L 1 density estimation using kernels, Ann. Statist., 20, 1658–1667.
De Angelis, D. and Young, G. A. (1992). Smoothing the bootstrap, Internat. Statist. Rev., 60, 45–56.
Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The L 1-View, Wiley, New York.
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multionmial estimator, Ann. Math. Statist., 27, 642–669.
Efron, B. (1979). Bootstrap methods: another look at the jackknife, Ann. Statist., 7, 1–26.
Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap, Chapman and Hall, New York.
Fernholz, L. T. (1983). von Mises Calculus for Statistical Functionals, Springer, New York.
Fernholz, L. T. (1991). Almost sure convergence of smoothed empirical distribution functions, Scand. J. Statist., 18, 255–262.
Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures, Ann. Probab., 18, 851–869.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, New York.
Hall, P. and Marron, J. S. (1987). Estimation of integrated squared density derivatives, Stat. Probab. Lett., 6, 109–115.
Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap, Ann. Statist., 17, 692–704.
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics. The Approach Based on Influence Functions, Wiley, New York.
Huber, P. J. (1981). Robust Statistics, Wiley, New York.
Lehmann, E. L. (1983). Theory of Point Estimation, Wiley, New York.
Marron, J. S. (1992). Bootstrap bandwidth selection. Exploring the Limits of Bootstrap (eds. R. Lepage and L. Billard), 249–262, Wiley, New York.
Natanson, I. P. (1961). Theory of Functions of a Real Variable, 1, Ungar, New York, rev. ed.
Parr, W. C. (1985). The bootstrap: some large sample theory and connections with robustness, Stat. Probab. Lett., 3, 97–100.
Prakasa Rao, B. L. S. (1983). Nonparametric Functional Estimation, Academic Press, New York.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.
Shao, J. (1989). Functional calculus and asymptotic theory for statistical analysis, Stat. Probab. Lett., 8, 397–405.
Sheather, S. J., Hettmansperger, T. P. and Donald, M. R. (1994). Data-based bandwidth selection for kernel estimators of the integral of f 2, Scand. J. Statist., 21, 265–275.
Silverman, B. W. (1981). Using kernel density estimates to investigate multimodality, J. Roy. Statist. Soc. Ser. B, 43, 97–99.
Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York.
Silverman, B. W. and Young, G. A. (1987). The bootstrap: to smooth or not to smooth? Biometrika, 74, 469–479.
Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap, Ann. Statist., 9, 1187–1195.
van der Vaart, A. (1994). Weak convergence of smoothed empirical processes, Scand. J. Statist., 21, 501–504.
von Mises, R. (1947). On the asymptotic distributions of differentiable statistical functions, Ann. Math. Statist., 18, 309–348.
Wang, S. (1989). On the bootstrap and smoothed bootstrap, Comm. Statist. Theory Methods, 18(11), 3949–3962.
Wu, T. J. (1995). Adaptive root n estimates of integrated squared density derivatives, Ann. Statist., 23, 1474–1495.
Young, G. A. (1990). Alternative smoothed bootstraps, J. Roy. Statist. Soc. Ser. B, 52(3), 477–484.
Author information
Authors and Affiliations
About this article
Cite this article
Cuevas, A., Romo, J. Differentiable Functionals and Smoothed Bootstrap. Annals of the Institute of Statistical Mathematics 49, 355–370 (1997). https://doi.org/10.1023/A:1003123215461
Issue Date:
DOI: https://doi.org/10.1023/A:1003123215461