On optimal adaptive estimation of a quadratic functional. (English) Zbl 0865.62024

Summary: Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule.
We investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known. It is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case. A lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case. On the other hand, we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case. Moreover, we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small.


62G07 Density estimation
62M20 Inference from stochastic processes and prediction
62M99 Inference from stochastic processes
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI


[1] BROWN, L. D. and LOW, M. G. 1996. A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. To appear. Z. · Zbl 0867.62023
[2] DONOHO, D. L., LIU, R. C. and MACGIBBON, 1990. Minimax risk over hy perrectangles, and implications. Ann. Statist. 18 1416 1437. Z. · Zbl 0705.62018
[3] DONOHO, D. L. and NUSSBAUM, M. 1990. Minimax quadratic estimation of a quadratic functional. J. Complexity 6 290 323. Z. · Zbl 0724.62039
[4] EFROMOVICH, S. 1994. On adaptive estimation of nonlinear functionals. Statist. Probab. Lett. 19 57 63. Z. · Zbl 0791.62041
[5] EFROMOVICH, S. and LOW, M. 1994. Adaptive estimates of linear functionals. Probab. Theory Related Fields 98 261 275. Z. · Zbl 0796.62037
[6] EFROMOVICH, S. and LOW, M. 1996. On Bickel and Ritov’s conjecture about adaptive estimation of some quadratic functionals. Ann. Statist. 24. To appear. Z. · Zbl 0859.62039
[7] EFROMOVICH, S. and PINSKER, M. S. 1984. Learning algorithm for nonparametric filtering. Automat. Remote Control 11 1434 1440. Z. · Zbl 0637.93069
[8] FAN, J. 1991. On the estimation of quadratic functionals. Ann. Statist. 19 1273 1294. Z. · Zbl 0729.62076
[9] HECKMAN, N. E. and WOODROOFE, M. 1991. Minimax Bay es estimation in nonparametric regression. Ann. Statist. 19 2003 2014. Z. · Zbl 0747.62014
[10] IBRAGIMOV, I. A. and KHAS’MINSKII, R. Z. 1980. Some estimation problems for stochastic differential equations. Lecture Notes in Control and Inform. Sci. B 25 1 12. Springer, Berlin. Z. · Zbl 0983.62052
[11] IBRAGIMOV, I. A., NEMIROVSKII, A. S. and KHAS’MINSKII, R. Z. 1986. Some problems on nonparametric estimation in Gaussian white noise. Theory Probab. Appl. 31 391 406. Z. · Zbl 0623.62028
[12] INGSTER, YU. I. 1986. Minimax testing of nonparametric hy potheses on a distribution density in the L -metrics. Theory Probab. Appl. 31 333 337. p Z. · Zbl 0629.62049
[13] LEPSKII, O. V. 1990. On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454 466. Z. · Zbl 0725.62075
[14] LEPSKII, O. V. 1992. On problems of adaptive estimation in white Gaussian noise. Advances in Soviet Mathematics 12 87 106. Z. · Zbl 0783.62061
[15] NUSSBAUM, M. 1994. Asy mptotic equivalence of density estimation and white noise. Technical Report, IAAS, Berlin. Z. Z PETROV, V. V. 1987. Limit Theorems for Sum of Independent Variables. Nauka, Moscow in. Russian.
[16] ALBUQUERQUE, NEW MEXICO 17131 PHILADELPHIA, PENNSy LVANIA 19104 E-MAIL: efrom@math.unm.edu E-MAIL: lowm@stat.wharton.upenn.edu
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.