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Question

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, \dots, x_n\} be the r.v. samples, drawn independently according to the true density p(x)p(x).

(a) Show that the mean of the estimator is

EX[p^(x)]=∫p(μ)k~(x−μ)dμ=p(x)∗k~(x),(5.1)\mathbb{E}_X [\hat{p}(x)] = \int p(\mu) \tilde{k}(x - \mu) d\mu = p(x) * \tilde{k}(x), \tag{5.1}

where ∗* is the convolution operator. What does this tell you about how the KDE is biased?