Empirical Processes: M estimation

over all (α, β). This gives us our standard least squares estimates (α̂, β̂), whose consistency we will take for granted in the ensuing discussion. Thus: (α̂, β̂) = argmin(α,β) Mn(α, β) . It is not difficult to check that: (α0, β0) = argmin(α,β)M(α, β) , where M(α, β) = P [(Y − α − β X)2] where P is the distribution of (X,Y ). Check that, up to a constant, M is a second order polynomial in (α, β) with a constant non–singular Hessian 2H, with h11 = 1, h22 = E(X2) and h12 = h21 = EX. We will apply the rate theorem, Theorem 3.2.5, to deduce the rate of convergence of (α̂, β̂). Firstly, as M(α, β)−M(α0, β0) = (α− α0, β − β0)H (α− α0, β − β0) where H is p.d., it is easy to see that the first condition of the theorem is satisfied, i.e. (α− α0, β − β0)H (α− α0, β − β0) ≥ constant× d((α, β), (α0, β0))