Download PDF by Bernard Moret, Henry D. Shapiro: Algorithms from P to NP, Vol. I: Design and Efficiency

By Bernard Moret, Henry D. Shapiro

ISBN-10: 0805380086

ISBN-13: 9780805380088

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Additional info for Algorithms from P to NP, Vol. I: Design and Efficiency

Example text

5 Exercises for Chapter 2 1. 1) for Hilbert spaces. 2. Let Z = (Z1 , . . , Zp )T be a p-dimensional multivariate normally distributed random vector with mean zero and covariance matrix Σp×p . We also write Z as the partitioned vector Z = (Y1T , Y2T )T , where Y1q×1 = (p−q)×1 (Z1 , . . , Zq )T and Y2 = (Zq+1 , . . , Zp )T , q < p, and ⎞ ⎛ q×(p−q) Σ12 Σq×q 11 ⎠, where Σ=⎝ (p−q)×q (p−q)×(p−q) Σ21 Σ22 Σ11 = E(Y1 Y1T ), Σ12 = E(Y1 Y2T ), Σ21 = E(Y2 Y1T ), Σ22 = E(Y2 Y2T ). Let H be the Hilbert space of all q-dimensional measurable functions of Z with mean zero, finite variance, and equipped with the covariance inner product.

It is instructive to consider the special case θ = (β T , η T )T . We first define the important notion of an efficient score vector and then show the relationship of the efficient score to the efficient influence function. Definition 8. , Seff (Z, θ0 ) = Sβ (Z, θ0 ) − Π(Sβ (Z, θ0 )|Λ). Recall that Π(Sβ (Z, θ0 )|Λ) = E(Sβ SηT ){E(Sη SηT )}−1 Sη (Z, θ0 ). Corollary 2. When the parameter θ can be partitioned as (β T , η T )T , where β is the parameter of interest and η is the nuisance parameter, then the efficient influence function can be written as T )}−1 {Seff (Z, θ0 )}.

Such estimators are referred to as super-efficient and for completeness we give the construction of such an estimator (Hodges) as an example. 1 Super-Efficiency Example Due to Hodges Let Z1 , . . , Zn be iid N (µ, 1), µ ∈ R. For this simple model, we know that the maximum likelihood estimator (MLE) of µ is given by the sample mean n Z¯n = n−1 i=1 Zi and that D(µ) n1/2 (Z¯n − µ) −−−→ N (0, 1). Now, consider the estimator µ ˆn given by Hodges in 1951 (see LeCam, 1953): Z¯n if |Z¯n | > n−1/4 µ ˆn = 0 if |Z¯n | ≤ n−1/4 .

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Algorithms from P to NP, Vol. I: Design and Efficiency by Bernard Moret, Henry D. Shapiro


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