Quadratic functional estimation in view of minimax goodness-of-fit testing from noisy data

We consider the convolution model $Y_i=X_i+ \varepsilon_i$, $i=1,\ldots,n$ of i.i.d. random variables $X_i$ having common unknown density $f$ are observed with an additive i.i.d. noise, independent of $X'$s. We assume that the density $f$ belongs to a smoothness class, has a characteristic function described either by a polynomial $|u|^{-\beta}$, $\beta >1/2$ (Sobolev class) or by an exponential $\exp(-\alpha |u|^r)$, $\alpha,~r>0$ (called supersmooth), as $|u| \to \infty$. The noise density is supposed to be known and such that its characteristic function decays either as $|u|^{-s}$, $s>0$ (polynomial noise) or as $\exp(-\gamma |u|^s)$, $s,~\gamma >0$ (exponential noise), as $|u| \to \infty$. We study the problems of estimating the quadratic functional $\int f^2$ and use this estimator for the goodness-of-fit test in $L_2$ distance, from noisy observations, in all possible combinations of the previous setups. We construct an estimator of $\int f^2$ based on the deconvolution kernel. 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 minimax upper bounds for the same estimator. For the goodness-of-fit test, we prove minimax upper bounds for a test statistic derived from the previous estimator. Surprisingly, in the case of supersmooth densities and polynomial noise we obtain parametric $n^{-1/2}$ minimax rate of testing. Finally, we give an approach unifying the proof of nonparametric minimax lower bounds. We prove them for Sobolev densities and polynomial noise, for Sobolev densities and exponential noise and for supersmooth densities with exponential noise such that $ r < s $. Note that in these last two setups we obtain exact testing constants associated to the asymptotic minimax rates