Constructing confidence intervals for selected parameters.

Area of Science:

• Statistical inference
• Bioinformatics
• High-dimensional data analysis

Background:

• Controlling the false coverage-statement rate (FCR) is crucial in large-scale statistical problems.
• Existing Benjamini and Hochberg (BY) confidence intervals (CIs) control FCR but are often uniformly inflated.
• There is a need for more efficient CIs in high-dimensional settings.

Purpose of the Study:

• To propose two novel methods for constructing shorter selective confidence intervals.
• To theoretically and numerically demonstrate the superiority of the proposed CIs over BY CIs.
• To apply the proposed methods to real-world data, such as microarray data.

Main Methods:

• Development of two distinct methods for constructing selective CIs.
• Method 1: Shortens CIs by allowing a reduced number of selected intervals.
• Method 2: Shortens CIs by permitting a proportion of intervals to cover uninteresting parameters.
• Theoretical analysis for asymptotic FCR control in independent data.

Main Results:

• The proposed selective CIs are theoretically proven to be uniformly shorter than BY CIs.
• Asymptotic control of FCR is demonstrated for independent data.
• Numerical simulations confirm theoretical findings and show effectiveness for correlated data.
• Successful application to HIV study microarray data highlights practical utility.

Conclusions:

• The proposed methods provide shorter selective CIs while maintaining FCR control.
• These methods offer a valuable alternative to standard BY CIs in large-scale statistical analyses.
• The approach is robust and applicable to both independent and correlated data, demonstrating practical advantages.