Let X1,..., Xn be an iid sample from a distribution F with density F' = f, and consider the KDE with a uniform kernel: X — βλία) - Σκ(Χ2). K(), Fr.h(2 1 nh K h { where K(t) = 11-1/2,1/2/(t), and h is called the bandwidth, and 1A(t) denotes the indicator function of a set A, i.e. 1 for xrEA 1A(t) 0 for 3A (Also note that K(t) is the density of the U(1-2, 2)) distribution.) Let Ph = : F(x + %) – F(= -) (a) Show that 27 h - 2 — 1 = = nhaph(1 – Pk). nh2Pu Ph E(fm,h(x)) and Var(fr,h(x)) h HINT: Notice that if Y1, ..., Yn are iid, then 21-11A(Y) has a binomial distri- bution. Why? (Think of Bernoulli trials...) Use this to find the distribution of E) 2!= 11-1/2,1/2 (***), and note that this equals nhfm,h().
