## Adaptive confidence interval for pointwise curve estimation.(English)Zbl 1106.62331

Summary: We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around $$f(x_0)$$, in either a white noise model or a regression setting. A suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor. The procedure does not require knowledge of the regularity of the unknown function $$f$$; it is also efficient for functions with a low degree of regularity.

### MSC:

 62G15 Nonparametric tolerance and confidence regions 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 62G07 Density estimation 62G08 Nonparametric regression and quantile regression
Full Text:

### References:

 [1] Arneodo, A., Jaffard, S., Levy-Vehel, J. and Meyer, Y. (1997). Méthodes d’ondelettes pour l’analyse de fonctions fractales et multifractales. Conférence CNRS ENS Cachan. [2] Bowman, A. W. and Hardle, W. (1988). Bootstrappingin nonparametric regression: Local adaptive smoothingand confidence bands. J. Amer. Statist. Assoc. 83 102-110. JSTOR: · Zbl 0644.62047 [3] C’irelson, B. S., Ibragimov, I. and Sudakov, V. (1976). Norm of Gaussian sample functions. In Proceedings of the 3rd Japan-URSS Symposium on Probability Theory (G. Maruyama amd J.V. Prohorov, eds.) Lecture Notes in Math. 550 20-41. Spring, New York. · Zbl 0359.60019 [4] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. · Zbl 0776.42018 [5] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032 [6] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR: · Zbl 0827.62035 [7] Faraway, T. (1990). Bootstrap selection of bandwidth and confidence bands for nonparametric regression. J. Statist. Comput. Simulation 37 37-44. · Zbl 0775.62092 [8] Feller, W. (1966). An introduction to Probability Theory and Its Applications. Wiley, New York. · Zbl 0138.10207 [9] Hall, P. (1991). Edgeworth expansions for nonparametric density estimators, with applications. Statistics 22 215-232. · Zbl 0809.62031 [10] Hall, P. (1992). Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann. Statist. 20 675-694. · Zbl 0748.62028 [11] Hall, P., Kerkyacharian, G. and Picard, D. (1998). Block threshold rules for curve estimation usingkernel and wavelet methods. Ann. Statist. 26 922-642. · Zbl 0929.62040 [12] Iouditski, A. and Lepskii, O. V. (1997). Personal communication. [13] Kerkyacharian, G., Picard, D. and Tribouley, K. (1996). Lp adaptive density estimation. Bernoulli 2 229-247. · Zbl 0858.62031 [14] Lepskii, O. V. (1990). On a problem of adaptive estimation in white Gaussian noise. Teor. Veoryatnost. i Primenen. 35 459-470 (in Russian). [Translated in (1991) Theory Probab. Appl. 35 454-466.] · Zbl 0725.62075 [15] Lepskii, O. V. (1991). Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682-697. · Zbl 0776.62039 [16] Lepskii, O. V., Mammen, E. and Spokoiny, V. G. (1994). Ideal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selection. Technical report. · Zbl 0885.62044 [17] Meyer, Y. (1990). Ondelettes. Hermann, Paris · Zbl 0735.42017 [18] Neumann, M. (1995). Automatic bandwidth choice and confidence intervals in nonparametric regression. Ann. Statist. 23 1937-1959. · Zbl 0856.62042 [19] Raimondo, M. (1998). Minimax estimation of sharp change points. Ann. Statist. 26 1379-1397. · Zbl 0929.62039 [20] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28-76. · Zbl 0798.60051
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.