Stat 300 A Theory of Statistics Homework 5 : Solutions

Abstract

We observe noisy measurements of this function yk = θ(kε) + zk, where (zk)k≤n ∼iid N(0, σ), and are interested in estimating θ with respect to the normalized square loss L(θ̂,θ) = ‖θ̂ − θ‖2/n. We define the discrete derivative by letting ∆θ(kε) = [θ((k + 1)ε)− θ(kε)]/ε for k ∈ {0, . . . , n− 2}, and ∆θ((n− 1)ε) = [θ(0)− θ((n− 1)ε)]/ε (periodic boundary conditions). We consider the following parameter class