Nonparametric Regression in Exponential Families
Lawrence
Brown, Tony Cai, and Harrison Zhou
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Abstract:
Most results in nonparametric regression theory are developed only for the
case of additive noise. In such a setting many smoothing techniques
including wavelet thresholding methods have been developed and shown to be
highly adaptive. In this paper we consider nonparametric regression in
exponential families which include, for example, Poisson regression,
binomial regression, and gamma regression. We propose a unified approach of
using a mean-matching variance stabilizing transformation to turn the
relatively complicated problem of nonparametric regression in exponential
families into a standard homoscedastic Gaussian regression problem. Then in
principle any good nonparametric Gaussian regression procedure can be
applied to the transformed data. In this paper we use a wavelet block
thresholding rule to construct the final estimator of the regression
function. The procedure is easily implementable. Both theoretical and
numerical properties of the estimator are investigated. The estimator
is shown to enjoy a high degree of adaptivity and spatial
adaptivity. It simultaneously attains the optimal rates of convergence under
integrated squared error over a wide range of Besov spaces and
achieves adaptive local minimax rate for estimating functions at a point.
The estimator also performs well numerically.
- Paper:
pdf file.
- Other related papers:
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