Adaptive estimation of a quadratic functional of a density by model selection

We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for U-statistics of order 2 due to Houdré and Reynaud.