On the Estimation of Quadratic Functionals

Abstract

We discuss the difficulties of estimating quadratic functionals based on observations $Y(t)$ from the white noise model $Y(t) = \\int^t_0 f(u) du + \\sigma W(t),\\quad t \\in \\lbrack 0, 1\\rbrack,$ where $W(t)$ is a standard Wiener process on $\\lbrack 0, 1\\rbrack$. The optimal rates of convergence (as $\\sigma \\rightarrow 0$) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating $\\int^1_0\\lbrack f^{(k)}(x)\\rbrack^2 dx$ under hyperrectangular constraints $\\sum = \\{f: |x_j(f)| \\leq Cj^{-\\alpha}\\}$ and weighted $l_p$-body constraints $\\sum_p = \\{f: \\sum^\\infty_1 j^r|x_j(f)|^p \\leq C\\}$ are computed explicitly, where $x_j(f)$ is the $j$th 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 $\\sigma \\rightarrow 0$) hypercube subproblems.