Laguerre estimation under constraint at a single point.

Summary This paper presents a general methodology for nonparametric estimation of a function s related to a nonnegative real random variable X, under a constraint of type s(0) = c. Three dierent examples are investigated: the direct observations model (X is observed), the multiplicative noise model (Y = XU is observed, with U following a uniform distribution) and the additive noise model (Y = X + V is observed where V is a nonnegative nuisance variable with known density). When a projection estimator of the target function is available, we explain how to modify it in order to obtain an estimator which satises the constraint. We extend risk bounds from the initial to the new estimator. Moreover if the previous estimator is adaptive in the sense that a model selection procedure is available to perform the squared bias/variance trade-o, we propose a new penalty also leading to an oracle type inequality for the new constrained estimator. The procedure is illustrated on simulated data, for density and survival function estimation.