On the Estimation of Quadratic Functionals

We discuss the difficulties of estimating quadratic functionals based on observations Y (t) from the white noise model Y (t) = Jf (u )du + cr W (t), t E [0,1], o where W (t) is a standard Wiener process on [0, 1]. The optimal rates of convergence (as cr -> 0) for estimating quadratic functionals under certain geometric constraints are 1 found. Specially, the optimal rates of estimating J[f (k)(x)f dx under hyperrectangular o constraints r = (J: Xj (f) ::; CFP ) and weighted lp -body constraints r p = (J: "Lj' IXj(f)IP ::; C) are computed explicitly, where Xj(f) is the jth Fourier1 Bessel coefficient of the unknown function f. We invent a new method for developing lower bounds based on testing two highly composite hypercubes, and address its advantages. The attainable lower bounds are found by applying the hardest I-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals (Le., the functionals which can be estimated at rate 0 (cr2», the difficulties of the estimation are captured by the hardest one dimensional subproblems and for estimating nonregular quadratic functionals (i.e. no 0 (cr1-consistent estimator exists), the difficulties are captured at certain finite dimensional (the dimension goes to infinite as cr -> 0) hypercube subproblems. AMS 1980 subject classificaJions. Primary 62COS; secondary 62M99