Vertically Weighted Averages in Hilbert Spaces and Applications to Imaging: Fixed-Sample Asymptotics and Efficient Sequential Two-Stage Estimation

Abstract

We discuss fixed-sample asymptotics and jackknife variance estimation for vertically weighted averages and the construction of related sequential two-stage confidence intervals. Those vertically weighted averages represent a class of nonlinear smoothers that are commonly applied to denoise observations without corrupting details (such as jumps in a time series or object boundaries in an image), detect those details, and design segmentation procedures. In addition to their extensive use in imaging, they have been also successfully applied in signal processing and financial data analysis. This article extends this approach to general functional data taking values in a Hilbert space and establishes the related asymptotic distribution theory in terms of central limit theorems and their sequential generalizations. In addition, focusing on real-valued data, the problems of variance estimation by the jackknife and two-stage estimation are studied. We show that the jackknife is consistent and asymptotically unbiased, thus providing an easy-to-use approach to evaluate the estimator's precision. Because the inhomogeneous variance of vertically weighted averages is a drawback when denoising data, we study the construction of fixed-width confidence intervals based on a two-stage sampling procedure in the spirit of Stein's (1945) seminal article. The proposed procedure can be shown to be consistent for the asymptotic optimal fixed-sample solution as well as asymptotically first-order efficient.

Keywords:

• Confidence intervals
• Edge detection
• Functional data
• Image processing
• Invariance principle
• Jackknife
• Resampling
• Sequential estimation

Subject Classifications:

• 60G10
• 62G08
• 62L12

ACKNOWLEDGMENTS

The author acknowledges helpful comments from an anonymous associate editor. Part of this article was written during a research stay at Seoul National University, Seoul, South Korea, and the author wants to thank Sangyeol Lee for the warm hospitality.

Notes

Recommended by Nitis Mukhopadhyay