Section: New Results

Local linear estimator of the conditional distribution function

Participants: S. Ferrigno, M. Maumy, A. Muller.

Consider $(X,Y)$, a random vector defined in $ℝ\times ℝ$. Here $Y$ is the variable of interest and $X$ the concomitant variable. As usual in the statistics literature, we work under the assumption that a sample $\left\{{(X_i,Y_i)}_{1\le i\le n}\right\}$ of independent and identically replica of $(X,Y)$ is available.

In order to explain the relationship between the variable of interest $Y$ and the factor $X$, the standard way is to rely on the regression function $E\left(Y\right|X=x)$. Because of numerous applications, the problem of estimating the regression function has been the subject of considerable interest during the last decades. However, it can be easily argued that the function $x\mapsto E\left(Y\right|X=x)$ alone does not capture the complexity of the relations between $X$ and $Y$.

In order to go one step further in this direction, we have chosen to work with another function. Namely, we study the conditional distribution function $F\left(y\right|X=x)=P(Y\le y|X=x)$ and a nonparametric estimator associated to this quantity. The distribution function has the advantage of completely characterizing the law of the random variable at stake, allowing to obtain the regression function, the density function, the moments and the quantile function. It should also be noticed that conditional distribution functions are used for the estimation of references curves in medical applications.

At a more technical level, our study is based on a local linear nonparametric estimator of the conditional distribution function instead of the widely spread Nadaraya-Watson estimator. Indeed, it is a well-known fact that the asymptotic bias of the Nadaraya-Watson estimator behaves somehow badly. Observe however that local polynomial techniques are good alternatives. Based on these techniques, here are the steps we have focused on in 2010-2011 :

• Our main result is the uniform law of the logarithm concerning the local linear estimator of the conditional distribution function (see [21] ). We investigate convergence in probability and almost sure convergence results.

• The uniform law of the logarithm has then been used to construct uniform asymptotic certainty bands for the conditional distribution function.

• The certainty bands alluded to above have been applied to simulated data.

• A variant of the test has been introduced in [20] .

Let us also mention that applications of these theoretical results to survival analysis are currently the object of active research.