Nonparametric Regression in Exponential Families

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.