Methods of Moments and Semiparametric Econometrics for Limited Dependent Variable Models pp 123-142 | Cite as

# Nonparametric Density Estimation

Chapter

## Abstract

In the linear model if we can estimate

*y*=*x′β + u*where*x*is a regressor vector and*E*(*u | x*)*=*0, we estimate*β*in*E*(*y*|*x*)*= x′β*However, the assumption of the linear model, or any nonlinear model for that matter, is a strong one. In nonparametric regression, we try to estimate*E*(*y*|*x*) without specifying the functional form. Since$$E(y{\kern 1pt} |{\kern 1pt} x) = \smallint y \cdot f(y|x)dy = \smallint y \cdot \{ f(y,x)/f(x)\} dy
$$

(1.1)

*f*(*y*,*x*) and*f*(*x*), we can also estimate*E*(*y*|*x*). In this chapter, we study nonparametric density estimation for*x*as a prelude to nonparametric regression in the next chapter. We will assume that*x*has a continuous density function*f (x)*If*x*is discrete, one can estimate*P*(*x*=*x*_{0}) either with the same estimation method used for the continuous case or with the number of observations with*x*_{ i }=*x*_{0}. There are several nonparametric density estimation methods available. The most popular is “kernel density estimation method” which we explore mainly. Other methods will be examined briefly in the last section. See Prakasa Rao (1983), Silverman (1986), Izenman (1991), Rosenblatt (1991) and Scott (1992) for more on nonparametric density estimation in general.## Keywords

Mean Square Error Kernel Density Estimation Nonparametric Regression Pointwise Convergence Kernel Estimator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1996