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Question

Problem 5.1 Bias and variance of the kernel density estimator​

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

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

Hint: the following properties will be helpful:

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

and Problem 1.4.