Skip to main content

Problem 5.1 Bias and variance of the kernel density estimator

In this problem, we will derive the bias and variance of the kernel density estimator. Let X={x1,⋯ ,xn}X = \{x_1, \cdots, x_n\} be the r.v. samples, drawn independently according to the true density p(x)p(x).

(b) Show that the variance of the estimator is bounded by

varX(p^(x))≤1nhdmax⁡x(k(x))E[p^(x)].(5.2)\mathrm{var}_X(\hat{p}(x)) \le \frac{1}{nh^d}\max_x(k(x))\mathbb{E}[\hat{p}(x)]. \quad (5.2)

Hint: the following properties will be helpful:

var(x)=E[x2]−(E[x])2≤E[x2],(5.3)\mathrm{var}(x) = \mathbb{E}[x^2] - (\mathbb{E}[x])^2 \le \mathbb{E}[x^2], \quad (5.3) k(x−xih)≤max⁡xk(x),(5.4)k\left(\frac{x - x_i}{h}\right) \le \max_x k(x), \quad (5.4)

and Problem 1.4.