Stein 1956: Efficient nonparametric testing and estimation

Under some regularity conditions (Stein refers to [33]), the maximum likelihood estimator θ̂n based on an i.i.d. sample X1, . . . ,Xn from pθ satisfies that √ n(θ̂n − θ) tends to a normal distribution with mean zero and variance ∇φ(θ)T I−1 θ ∇φ(θ), and hence attains this bound. Even if the parameter set may be multi-dimensional, this lower bound for estimation of a real-valued parameter φ(θ) can already be obtained from considering a one-dimensional