ON FINITE DIMENSIONAL APPROXIMATION IN NONPARAMETRIC REGRESSION

無母數迴歸之有限維空間逼近論

魏文翔(Wen Hsiang Wei)

《中國統計學報》 54卷4期 (2016/12) Pp. 186-204

摘要

For the function from a real separable Banach space into a real separable Banach space, i.e., a possibly nonlinear operator, in nonparametric regression, theoretical results are established for the estimator based on finite dimensional approximation. A new concept ＂approximatability＂ is presented and the operators of interest are proved to be approximatable under different situations. The results concerning both consistency and weak convergence of the estimator are obtained. Statistical applications of these theoretical results are given.

關鍵字

Approximatability ； Banach spaces ； Consistency ； Nonlinear operator ； Nonparametric regression ； Weak convergence

並列摘要

本文建立針對無母數迴歸(nonparametric regression)中被估計函數為一定義及取值皆在可分離之實數場巴拿赫空間(real separable Banach space)的非線性算子(nonlinear operator)而其估計算子所形成空間為有限維情形下發展相關理論結果。在估計過程中，延伸出一新的概念，稱之為可逼近性(approximatability)。被估計算子在不同狀況下是可逼近的理論結果被證明,而估計算子之一致性(consistency)及弱收斂性(weak convergence)亦被證明。此外，這些理論能應用到不同的統計模式。

並列關鍵字

可逼近性 ； 巴拿赫空間(Banach space) ； 一致性 ； 非線性算子 ； 無母數迴歸 ； 弱收斂性

參考文獻

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