Abstract
In this paper we are interested in approximating the conditional expectation of a given random variable X with respect to the standard normal distribution N(0, 1). Actually we have shown that the conditional expectation E(X|Z) could be interpolated by an N degree polynomial function of Z, φN(Z) where N is the number of observations recorded for the conditional expectation E(X|Z = z). A pointwise error estimation has been proved under reasonable condition on the random variable X.
Highlights
Since the outstanding Stone-Weierstrass result about the approximation of continuous functions by polynomials on compact sets, (Rudin, 1976) and (Prolla, 1993), polynomial interpolation has become the corner stone of numerical analysis and approximation theory
In this paper we are interested in approximating the conditional expectation of a given random variable X with respect to the standard normal distribution N(0, 1)
We have shown that the conditional expectation E(X|Z) could be interpolated by an N degree polynomial function of Z, φN(Z) where N is the number of observations recorded for the conditional expectation E(X|Z = z)
Summary
Since the outstanding Stone-Weierstrass result about the approximation of continuous functions by polynomials on compact sets, (Rudin, 1976) and (Prolla, 1993), polynomial interpolation has become the corner stone of numerical analysis and approximation theory. This could be done if we consider the conditional expectation of a given random variable X with respect to another known one Z This is because this latter can be expressed as E(X|Z) = φ(Z) where φ is L2 with respect to the probability distribution dμZ of Z when X is of finite variance, the function φ is called regression function. In the present paper our main interest is to interpolate the conditional expectation random variable E(X|Z) with a polynomial function of Z, where X is supposed to have finite variance and Z is a random variable that follows the standard normal distribution N(0, 1). We have shown the convergence of the polynomial sequence φN pointwisely and uniformly to the function E(X|Z = ·) = φ(·), which leads to the existence of a continuous version for the function φ, and so we have answered to questions 1-3 at the same time. The last section is devoted to the numerical illustration of our result where we have tested the interpolation algorithm and used it to simulate the conditional expectation E(X|Z)
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