Supplementary Material: Proofs of Theorems 1–4, Figures 1–5, Simulations 2–3 and Tables 2–3
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چکیده
Then the first-order derivative of f(cj(Θ0; y)) with respect to cj(Θ0; y) is: − 1 n ∑n i=1 1 νj0 φ( cj(Θ0;y)−Wij(Θ0) νj0 ) < 0. Thus it ensures uniqueness of the root of f(cj(Θ0; y)) = 0 in the entire domain of cj(y). Note that, ĉj(Θ0; y) satisfied (S.1), hence f(ĉj(Θ0; y)) = 0. On the other hand, f(cj0(y)) → 0 (given the uniform law of large numbers). By the standard inverse function theorem, we have that ĉj (Θ0; y) → cj0(y). Step 2: Show Θ̂ → Θ0. ∗School of Statistics, Southwestern University of Finance and Economics, Chengdu, China †Department of Statistics, The Chinese University of Hong Kong, Hong Kong ‡Department of Biostatistics, University of Michigan, USA
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