2Nonparametric Regression
2.1 Introduction
Consider the bivariate random variable
where X denotes an explanatory variable and Y denotes the response or the dependent variable. Consider the observations
on the pair (X, Y). Although in this discussion, we let X be a scalar, the ideas presented here can easily be generalized to the multidimensional case, i.e., when
. Similarly, the response variable Y is in
, though multivariate regression is also possible to consider. Moreover, let Y be continuous and in the sequel we will impose some further moment conditions. Nonparametric regression is concerned with the situation when the regression function, i.e., the conditional expected value of Y given X has an arbitrary shape, apart from satisfying some smoothness conditions.
Specifically, our interest lies in estimating the function m, which is the nonparametric regression function
where denotes the conditional expectation of Y given X = x. For simplicity of notation, we write ...