ON THE ESTIMATION OF QUADRATIC FUNCTIONALS

• 成果类型: Article
• 署名作者: FAN, JQ
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176348249
• 发表日期: 1991
• 页码: 1273-1294
• 关键词: squared density derivatives; regression-models; CONVERGENCE; rates

摘要

We discuss the difficulties of estimating quadratic functionals based on observations Y(t) from the white noise model Y(t) = integral 0t(f)(u) du + sigma-W(t), t is-an-element-of [0, 1], where W(t) is a standard Wiener process on [0, 1]. The optimal rates of convergence (as sigma --> 0) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating integral 0(1)[f(k)(x)]2 dx under hyperrectangular constraints SIGMA = {f: \x(j)(f)\ less-than-or-equal-to Cj-alpha} and weighted l(p)-body constraints SIGMA-p = {f: SIGMA-1 infinity j(r)\x(j)(f)\p less-than-or-equal-to C} are computed explicitly, where x(j)(f) is the jth Fourier-Bessel coefficient of the unknown function f. We develop lower bounds based on testing two highly composite hypercubes and address their advantages. The attainable lower bounds are found by applying the hardest one-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals [i.e., the functionals which can be estimated at rate O(sigma-2)], the difficulties of the estimation are captured by the hardest one-dimensional subproblems, and for estimating nonregular quadratic functionals [i.e., no O(sigma-2)-consistent estimator exists], the difficulties are captured at certain finite-dimensional (the dimension goes to infinity as sigma --> 0) hypercube subproblems.