8 research outputs found
ํ๋ฒํธ ๋ฐ์๋ณ์๋ฅผ ๊ฐ๋ ๊ฐ๋ฒ ํ๊ท
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ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ์์ฐ๊ณผํ๋ํ ํต๊ณํ๊ณผ, 2018. 8. ๋ฐ๋ณ์ฑ.This paper develops a foundation of methodology and theory for the estimation of structured nonparametric regression models with Hilbertian responses. Our method and theory are focused on the additive model, while the main ideas may be adapted to other structured models. For this, the notion of Bochner integration is introduced for Banach-space-valued maps as a generalization of Lebesgue integration. Several statistical properties of Bochner integrals, relevant for our method and theory, and also of importance in their own right, are presented for the first time. Our theory is complete. The existence of our estimators and the convergence
of a practical algorithm that evaluates the estimators are established. These results are non-asymptotic as well as asymptotic. Furthermore, it is proved that the estimators of component maps achieve the univariate error rates in pointwise, and uniform convergence, and converge jointly in distribution to Gaussian random
elements. Our numerical examples include the cases of functional, density-valued and
simplex-valued responses, which demonstrate the validity of our approach.1 Introduction
2 Bochner Smooth Backfitting
2.1 Examples of Hilbertian response
2.2 Bochner integration
2.3 Lebesgue-Bochner spaces of additive maps
2.4 Bochner integral equations and backfitting algorithm
2.5 Practical implementation
3 Existence and Algorithm Convergence
3.1 Projection operators
3.2 Compactness of projection operators
3.3 Existence of B-SBF estimators
3.4 Convergence of B-SBF algorithm
4 Asymptotic properties
4.1 Rates of convergence
4.2 Asymptotic distribution and asymptotic independence
5 Numerical Study
5.1 Bandwidth selection
5.2 Simulation study with density response
5.3 Real data analysis with functional response
5.4 Real data analysis with simplex-valued response
6 Appendix (Additional Results and Selected Proofs)
6.1 Lemmas and additional propositions
6.2 Proof of Theorem 3.2.1
6.3 Proof of Theorem 3.2.2
6.4 Proof of Theorem 3.4.2
6.5 Proof of Theorem 4.1.1
6.6 Proof of Theorem 4.2.1
6.7 Proof of Theorem 4.2.2
6.8 Proof of Lemma 6.1.1
6.9 Proof of Lemma 6.1.2
6.10 Proof of Lemma 6.1.3
6.11 Proof of Lemma 6.1.4
6.12 Proof of Lemma 6.1.6
6.13 Proof of Proposition 6.1.2
Abstract (in Korean)Docto