1.3. The Cram´ er-Rao Lower Bound 7 The Fisher information for a statistical experiment of size n is the vari- ance of the total Fisher score function, In(θ) = Varθ L n (θ) = Eθ ( L n (θ) ) 2 = Eθ ∂ ln p (X1,...,Xn, θ) ∂ θ 2 = Rn (∂ p (x1,...,xn,θ)/∂θ)2 p (x1,...,xn,θ) dx1 . . . dxn . Lemma 1.7. For independent observations, the Fisher information is ad- ditive. In particular, for any θ ∈ Θ , the equation holds In(θ) = n I(θ). Proof. As the variance of the sum of n independent random variables, In(θ) = Varθ L n (θ) = Varθ l (X1 , θ) + . . . + l (Xn , θ) = n Varθ l (X1 , θ) = n I(θ). In view of this lemma, we use the following definition of the Fisher information for a random sample of size n: In(θ) = n Eθ ∂ ln p (X, θ) ∂ θ 2 . Another way of computing the Fisher information is presented in Exercise 1.1. 1.3. The Cram´ er-Rao Lower Bound A statistical experiment is called regular if its Fisher information is con- tinuous, strictly positive, and bounded for all θ ∈ Θ . Next we present an inequality for the variance of any estimator of θ in a regular experiment. This inequality is termed the Cram´ er-Rao inequality, and the lower bound is known as the Cram´ er-Rao lower bound. Theorem 1.8. Consider an estimator ˆ n = ˆ n (X1,...,Xn) of the parame- ter θ in a regular experiment. Suppose its bias bn(θ) = Eθ ˆ n − θ is con- tinuously differentiable. Let b n (θ) denote the derivative of the bias. Then the variance of ˆ n satisfies the inequality (1.1) Varθ ˆ n ≥ ( 1 + b n (θ) )2 In(θ) , θ ∈ Θ. Proof. By the definition of the bias, we have that θ + bn(θ) = Eθ ˆ n = Rn ˆ n (x1,...,xn) p (x1,...,xn,θ) dx1 . . . dxn.

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