Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62F12 Asymptotic properties of estimators
- 62G05 Estimation
- 62G10 Hypothesis testing
- 62G20 Asymptotic properties

**Résumé:** 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.

**Mots Clés:** *Asymptotic efficiency ; convolution model ; exact constant in nonparametric tests ; goodness-of-fit tests ; infinitely differentiable functions ; quadratic
functional estimation ; minimax tests ; Sobolev classes*

**Date:** 2004-09-27

**Prépublication numéro:** *PMA-936*