10. (Haar density Estimation.) Let X1,…,Xn ~ f for some density f on [0, 1]. Let’s consider…
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10. (Haar density Estimation.) Let X1,…,Xn ~ f for some density f on [0, 1]. Let’s consider constructing a wavelet histogram. Let f and ? be the Haar father and mother wavelet. Write f(x) ˜ f(x) + J -1 j=0 2 j-1 k=0 ßj,k?j,k(x) where J ˜ log2(n). Let ß j,k = 1 n n i=1 ?j,k(Xi). 348 21. Smoothing Using Orthogonal Functions (a) Show that ß j,k is an unbiased estimate of ßj,k. (b) Define the Haar histogram f (x) = f(x) + B j=0 2 j-1 k=0 ß j,k?j,k(x) for 0 = B = J – 1. (c) Find an approximate expression for the MSE as a function of B. (d) Generate n = 1, 000 observations from a Beta (15,4) density. Estimate the density using the Haar histogram. Use leave-one-out cross validation to choose B
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