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4) imply E{gn (t) − g(t)}2 = σ02 + O(µ2n ) + o(νn−1 ) + o(µ2n ). 5) not only provides an optimum convergence rate, but proposes a theoretical selection for the smoothing parameter involved in the (1) (2) weight functions ωni as well. 2, we have E{gn (t) − g(t)}2 = σ02 4 + O(h4n ) + o(n−1 h−1 n ) + o(hn ). 6) This suggests that the theoretical selection of hn is proportional to n−1/5 . 4. 7) 48 3. ESTIMATION OF THE NONPARAMETRIC COMPONENT uniformly over t ∈ [0, 1], where sn → 0 as n → ∞. d. 7) is reasonable.

N. √ ∗ The bootstrap principle asserts that the distributions of n(βLS − βLS ) and n(σn2∗ − σn2 ), which can be computed directly from the data, can approximate √ √ the distributions of n(βLS − β) and n(σn2 − σ 2 ), respectively. As shown later, this approximation works very well as n → ∞. The main result of this section is given in the following theorem. 3 hold. If max1≤i≤n ui ≤ C0 < ∞ and Eε41 < ∞. 1) and √ √ supx P ∗ { n(σn2∗ − σn2 ) < x} − P { n(σn2 − σ 2 ) < x} → 0 where P ∗ denotes the conditional probability given Y.

1≤i≤n Proof. Note that n ωnj (Wi )(Yj − XjT βLS )2 Hn (Wi ) = j=1 n ωnj (Wi ){XjT (β − βLS ) + g(Ti ) + εi }2 = j=1 n = (β − βLS )T n j=1 j=1 n n ωnj (Wi )ε2i + 2 + ωnj (Wi )g 2 (Ti ) ωnj (Wi )Xj XjT (β − βLS ) + j=1 n ωnj (Wi )XjT (β − βLS )g(Ti ) j=1 n ωnj (Wi )XjT (β − βLS )εi + 2 +2 j=1 ωnj (Wi )g(Ti )εi . 7) is therefore OP (n−2/3 ) since n j=1 Xj XjT is a symmetric matrix, 0 < ωnj (Wi ) ≤ Cn−2/3 , n {ωnj (Wi ) − Cn−2/3 }Xj XjT j=1 is a p × p nonpositive matrix, and βLS − β = OP (n−1/2 ). 7) is easily shown to be of order OP (n1/3 c2n ).

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Accuracy of MSI testing in predicting germline mutations of MSH2 and MLH1 a case study in Bayesian m by Chen S.

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