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Goodness-of-fit testing and quadratic functional estimation from indirect observations. (English) Zbl 1126.62028

Summary: We consider the convolution model where i.i.d. random variables \(X_i\) having unknown density \(f\) are observed with additive i.i.d. noise, independent of the \(X_i\)’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 \(\int f^{2}\) 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.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
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