The Annals of Statistics

On optimal adaptive estimation of a quadratic functional

Sam Efromovich and Mark Low

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Abstract

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.

Article information

Source
Ann. Statist., Volume 24, Number 3 (1996), 1106-1125.

Dates
First available in Project Euclid: 20 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032526959

Digital Object Identifier
doi:10.1214/aos/1032526959

Mathematical Reviews number (MathSciNet)
MR1401840

Zentralblatt MATH identifier
0865.62024

Subjects
Primary: 62C05: General considerations
Secondary: 62E20: Asymptotic distribution theory 62J02: General nonlinear regression 62G04 62M99: None of the above, but in this section

Keywords
Functional estimation adaptation optimal risk convergence filtering nonparametric function

Citation

Efromovich, Sam; Low, Mark. On optimal adaptive estimation of a quadratic functional. Ann. Statist. 24 (1996), no. 3, 1106--1125. doi:10.1214/aos/1032526959. https://projecteuclid.org/euclid.aos/1032526959


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  • ALBUQUERQUE, NEW MEXICO 17131 PHILADELPHIA, PENNSy LVANIA 19104 E-MAIL: efrom@math.unm.edu E-MAIL: lowm@stat.wharton.upenn.edu