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stat612-notes2-mod

# stat612-notes2-mod - Statistics 612 Regular Parametric...

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Statistics 612: Regular Parametric Models and Likelihood Based Inference Moulinath Banerjee March 30, 2009 We continue our discussion of likelihood based inference for parametric models; in particular, we will talk more about information bounds in the context of parametric models, and the role they play in likelihood based inference. We first introduce the multiparameter version of the celebrated Cramer-Rao inequality. I will not describe the underlying assumptions in details. These are the usual sorts of assumptions one makes for parametric models, in order to be able to establish sensible results. See Page 11 of Chapter 3 of Wellner’s notes for a detailed description of the conditions involved. For a multidimensional parametric model { p ( x, θ ) : θ Θ R k } , the information matrix I ( θ ) is given by: I ( θ ) = E θ ( ˙ l ( X, θ ) , ˙ l ( X, θ ) T ) = - E θ ¨ l ( X, θ ) , where ˙ l ( X, θ ) = ∂ θ l ( X, θ ) being a k × 1 column vector (recall that l ( x, θ ) = log p ( x, θ )), and ¨ l ( x, θ ) = 2 ∂ θ ∂ θ T l ( X, θ ) , is a k × k matrix. Consider a smooth real-valued function q ( θ ) that is estimated by some statistic T ( X ), and let ˙ q ( θ ) denote the derivative of q (written as a k × 1 vector). Let b ( θ ) = E θ ( T ( X )) - q ( θ ) be the bias of the estimator T , and let ˙ b ( θ ) denote the derivative of the bias. We then have: Var θ ( T ( X )) ( ˙ q ( θ ) + ˙ b ( θ )) T I - 1 ( θ ) ( ˙ q ( θ ) + ˙ b ( θ )) . In particular, if T ( X ) is unbiased for q ( θ ), then Var θ ( T ( X )) ˙ q ( θ ) T I - 1 ( θ ) ˙ q ( θ ) . For a proof of this result, see Page 12 of Chapter 3 of Wellner’s notes – the proof runs along lines similar to the one–dimensional case. We will not be worried about the construction of exact 1

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unbiased estimators for q ( θ ) that attain the information bound; in the vast majority of situations this is not feasible. Rather, we focus on the connection of the MLE ˆ θ n to the information bound arising from the multiparameter inequality above. Consider the asymptotically linear representation of the MLE given by: n ( ˆ θ n - θ ) = 1 n n i =1 I ( θ ) - 1 ˙ l ( X i , θ ) + o p (1) . Invoke the Delta method to obtain: n ( q ( ˆ θ n ) - q ( θ )) = 1 n n i =1 ˙ q ( θ ) T I ( θ ) - 1 ˙ l ( X i , θ ) + o p (1) . It is easily seen that the asymptotic variance of n ( q ( ˆ θ n ) - q ( θ )) is exactly ˙ q ( θ ) T I - 1 ( θ ) q ( θ ), the information bound arising from the multiparameter Cramer Rao inequality. The function ˙ q ( θ ) T I ( θ ) - 1 ˙ l ( x, θ ) (that provides a linearization of the MLE) is called the efficient influence function for estimating q ( θ ). Motivated by the above considerations, we define efficient influence functions and information bounds for vector-valued functions of θ . Let ν be a Euclidean parameter defined on a regular parametric model and P = { P θ : θ Θ } . We can identify ν with the parametric function q : Θ R m defined by: q ( θ ) = ν ( P θ ) , for P θ ∈ P .
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stat612-notes2-mod - Statistics 612 Regular Parametric...

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