On the approximation of a conditional expectation

In this paper, we discuss how to approximate the conditional expectation of a random variable Y given a random variable X, i.e. E (Y|X). We propose and compare two different non parametric methodologies to approximate E (Y|X). The first approach (namely the OLP method) is based on a suitable approximation of the sigma-algebra generated by X. A second procedure is based on the well known kernel non-parametric regression method. We analyze the convergence properties of the OLP estimator and we compare the two approaches with a simulation study.