Goodness-of-fit testing and quadratic functional estimation from indirect observations

Abstract

We consider the convolution model where i.i.d. random variables Xi having unknown density f are observed with additive i.i.d. noise, independent of the X's. We assume that the density f belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically.We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional f 2 based on indirect observations. When the unknown density is smoother enough than the noise density, we prove that this estimator is n −1/2 consistent, asymptotically normal and efficient (for the variance we compute). Otherwise, we give nonparametric upper bounds for the risk of the same estimator.We give an approach unifying the proof of nonparametric minimax lower bounds for both problems. We establish them for Sobolev densities and for supersmooth densities less smooth than exponential noise. In the two setups we obtain exact testing constants associated with the asymptotic minimax rates.

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

“…Our rates of convergence are similar to those determined in (Butucea 2007, Theorem 4) which considers the density convolution, a non-adaptive estimator based on kernel, and the risk:…”

“…To the best of our knowledge, the estimation of f 2 2 from a convolution problem has been firstly studied in Butucea (2007). The model considered is different to (1) and can be described as follows: i.i.d.…”

“…To reach this goal, we use wavelet estimators. As in (Butucea 2007, Theorem 4), we distinguish two cases according to the nature of g: the ordinary smooth case where the Fourier coefficients of g decrease in a polynomial fashion, and the supersmooth case where they decrease in an exponential fashion.…”

Comments on: An updated review of Goodness-of-Fit tests for regression models

“…Some results are available for related indirect models. Here tests for the parametric form of the density in deconvolution problems have been discussed in Butucea (2007), and testing parametric model assumptions in the presence of instrumental variables (which is closely related to statistical inverse problems) has been considered in Holzmann (2007). In order to illustrate the difficulties which appear while testing model assumptions in inverse regression models consider the problem of testing for a parametric form of the function m, where the operator K is a convolution defined by…”

Nonparametric specification tests for stochastic volatility models based on volatility density

“…Nonparametric specification tests for diffusion processes were investigated by Aït-Sahalia (1996), Hong and Li (2005), Corradi and Swanson (2005), Chen et al (2008), Aït-Sahalia et al (2010) and Aït-Sahalia and Park (2012), among others. Specification tests involving a deconvolution kernel estimator is relatively new, see Butucea (2007) and Holzmann et al (2007).…”