Simultaneous confidence region of an embedded one-dimensional curve in multi-dimensional space

This paper focuses on the simultaneous confidence region of a one-dimensional curve embedded in multi-dimensional space. Local linear regression is applied component-wise to each variable in multi-dimensional data, which yields an estimator of the one-dimensional curve. A simultaneous confidence region of the curve is proposed based on this estimator and theoretical results for the estimator and the region are developed under some reasonable assumptions. Practically efficient algorithms to determine the thickness of the region are also addressed. The effectiveness of the region is investigated through simulation studies and applications to artificial and real datasets, which reveal that the proposed simultaneous confidence region works well.