This thesis deals with minimax rates of convergence for estimation of density functions on the real line. The densities are assumed to be location mixtures of normals, a global regularity requirement that creates subtle difficulties for the application of standard minimax lower bound techniques. The subtlety is reflected in the lower bounds under various loss functions, which are shown to be slower than the parametric rate inflated by only logarithmic factors. The sharpest results are obtained for squared error loss where the optimal inflation factor is the square root of the logarithm of the sample size. All the lower bounds are developed using novel Fourier and orthogonal polynomial methods.
|Advisor:||Pollard, David B., Zhou, Harrison H.|
|School Location:||United States -- Connecticut|
|Source:||DAI-B 73/12(E), Dissertation Abstracts International|
|Keywords:||Hermite polynomials, Legendre polynomials, Minimax bounds, Minimax rates, Mixture normals, Normal location mixtures|
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