Minimum Area Confidence Set Optimality for Simultaneous Confidence Bands for Percentiles in Linear Regression

TL;DR

This paper introduces the Minimum Area Confidence Set criterion for optimal simultaneous confidence bands in linear regression, demonstrating that asymmetric bands often outperform symmetric ones in size, with a new efficient computation method and real data application.

Contribution

It proposes a novel MACS criterion for comparing and constructing optimal SCBs for percentiles in linear regression, emphasizing asymmetric bands and computational efficiency.

Findings

Asymmetric SCBs have substantially smaller confidence set areas than symmetric ones.

The MACS criterion consistently favors asymmetric SCBs as optimal.

A new efficient method for calculating critical constants of SCBs is developed.

Abstract

Simultaneous confidence bands (SCBs) for percentiles in linear regression are valuable tools with many applications. In this paper, we propose a novel criterion for comparing SCBs for percentiles, termed the Minimum Area Confidence Set (MACS) criterion. This criterion utilizes the area of the confidence set for the pivotal quantities, which are generated from the confidence set of the unknown parameters. Subsequently, we employ the MACS criterion to construct exact SCBs over any finite covariate intervals and to compare multiple SCBs of different forms. This approach can be used to determine the optimal SCBs. It is discovered that the area of the confidence set for the pivotal quantities of an asymmetric SCB is uniformly and can be very substantially smaller than that of the corresponding symmetric SCB. Therefore, under the MACS criterion, exact asymmetric SCBs should always be preferred. Furthermore, a new computationally efficient method is proposed to calculate the critical constants of exact SCBs for percentiles. A real data example on drug stability study is provided for illustration.