and n1/2

In this lecture, we discuss kernel estimation of probability density functions (PDF). Nonparametric density estimation is one of the central problems in statistics. In economics, nonparametric density estimation plays important roles in various areas such as, for example, industrial organization (Guerre, Perrigne, and Vuong, 2000), empirical finance (Ait-Sahalia, 1996), and etc. These notes borrow from the following sources: Li and Racine (2007), Pagan and Ullah (1999), and Härdle and Linton (1994). Kernel density estimation Assumption 1 (a) Suppose {Xi: i = 1,..., n} is a collection of iid random variables drawn from a distri-bution with the CDF F and PDF f. (b) In the neighborhood Nx of x, f is bounded and twice continuously differentiable with bounded derivatives. (When discussing f (x), we will implicitly assume that f (x) exists at x.) The econometrician’s objective is to estimate f without imposing any functional form (parametric) assumptions on the PDF. First, consider estimation of F. Since F (x) = E1 {Xi ≤ x}, an estimator of F is given b