Jackknife variance estimation for common mean estimators under ordered variances and general two-sample statistics

TL;DR

This paper develops a jackknife variance estimation method for common mean estimators with ordered variances, providing theoretical consistency, a CLT, and practical confidence interval assessments for complex two-sample statistics.

Contribution

It introduces a jackknife-based variance estimation approach for common mean estimators under ordered variances, extending to general two-sample statistics with theoretical guarantees.

Findings

Jackknife estimators are consistent for the class of two-sample statistics considered.

A central limit theorem is established for the common mean estimators.

Simulation studies demonstrate the accuracy of confidence intervals based on the proposed method.

Abstract

Samples with a common mean but possibly different, ordered variances arise in various fields such as interlaboratory experiments, field studies or the analysis of sensor data. Estimators for the common mean under ordered variances typically employ random weights, which depend on the sample means and the unbiased variance estimators. They take different forms when the sample estimators are in agreement with the order constraints or not, which complicates even basic analyses such as estimating their variance. We propose to use the jackknife, whose consistency is established for general smooth two--sample statistics induced by continuously G\^ateux or Fr\'echet differentiable functionals, and, more generally, asymptotically linear two--sample statistics, allowing us to study a large class of common mean estimators. Further, it is shown that the common mean estimators under consideration satisfy a central limit theorem (CLT). We investigate the accuracy of the resulting confidence intervals by simulations and illustrate the approach by analyzing several data sets.