On the estimation of quadratic functionals

Jianqing Fan

Statistics Theory and Methods mathscidoc:1912.43301

The Annals of Statistics, 19, (3), 1273-1294, 1991
We discuss the difficulties of estimating quadratic functionals based on observations Y (t) from the white noise model Y (t) where Y (t) is a standard Wiener process on Y (t) . The optimal rates of convergence (as Y (t) ) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating Y (t) under hyperrectangular constraints Y (t) and weighted Y (t) -body constraints Y (t) are computed explicitly, where Y (t) is the Y (t) th Fourier-Bessel coefficient of the unknown function Y (t) . 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 [ie, the functionals which can be estimated
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  title={On the estimation of quadratic functionals},
  author={Jianqing Fan},
  booktitle={The Annals of Statistics},
Jianqing Fan. On the estimation of quadratic functionals. 1991. Vol. 19. In The Annals of Statistics. pp.1273-1294. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221113558681738861.
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