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Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. (English) Zbl 0722.62030

The paper is concerned with the clever idea of using bootstrap samples of essentially smaller size \(n_ 1\) than the size of the original sample n. More precisely, the author considers a bootstrap version \(f^*(\cdot | n_ 1,h_ 1)\) of the kernel density estimate \(\hat f(\cdot | n,h)\) and proves, in particular, that quantities like \[ (1)\quad E[\hat f(x| n_ 1,h_ 1)-f(x)]^ p,\text{ and } (2)\quad E[\hat f^*(x| n_ 1,h_ 1)-\hat f(x| n,h)]^ p \] (which obviously take account of both variance and bias of \(\hat f\) and \(\hat f^*)\) are close to each other for \(n_ 1<cn^{1-\delta}.\)
The same statment concerning integral (in x) versions of (1) and (2) is proved. Some emphasis is on the problem of bootstrap estimation of a bias.

MSC:

62G09 Nonparametric statistical resampling methods
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Burkholder, D. L., Distribution function inequalities for martingales, Ann. Probab., 1, 19-42 (1973) · Zbl 0301.60035
[2] Csörgő, S.; Deheuvels, P.; Mason, D., Kernel estimates of the tail index of a distribution, Ann. Statist., 13, 1050-1077 (1985) · Zbl 0588.62051
[3] Devroye, L.; Györfi, L., (Nonparametric Density Estimation: The \(L_1\) View (1984), Wiley: Wiley New York)
[4] Farraway, J.; Jhun, M., (Bootstrap Choice of Bandwidth for Density Estimation (1987), Department of Statistics, University of Michigan), Tech. Report No. 157
[5] Hall, P., On some simple estimates of an exponent of regular variation, J. Roy. Statist. Soc. Ser. B, 44, 37-42 (1982) · Zbl 0521.62024
[6] Hall, P.; Marron, J. S., Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation, Probab. Theory Related Fields, 74, 567-581 (1987) · Zbl 0588.62052
[7] Hill, B. M., A simple approach to inference about the tail of a distribution, Ann. Statist., 3, 1163-1174 (1975) · Zbl 0323.62033
[8] Petrov, V. V., (Sums of Independent Random Variables (1975), Springer: Springer Berlin) · Zbl 0322.60043
[9] Prakasa Rao, B. L.S, (Nonparametric Functional Estimation (1975), Academic: Academic New York)
[10] Silverman, B. W., (Density Estimation for Statistics and Data Analysis (1986), Chapman & Hall: Chapman & Hall London) · Zbl 0617.62042
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