Confidence Regions Near Singular Information and Boundary Points With Applications to Mixed Models

TL;DR

This paper develops confidence regions for parameters with potentially singular Fisher information matrices, especially near boundaries, ensuring reliable inference in mixed models with small sample sizes.

Contribution

It introduces a method to construct confidence regions with correct coverage near singular information points, extending previous scalar-only results to vector parameters.

Findings

Confidence regions achieve near-nominal coverage with small samples.

Method applies to parameters close to boundary points, including zero in mixed models.

Test statistic maintains chi-square distribution asymptotically, even at boundary points.

Abstract

We propose confidence regions with asymptotically correct uniform coverage probability of parameters whose Fisher information matrix can be singular at important points of the parameter set. Our work is motivated by the need for reliable inference on scale parameters close or equal to zero in mixed models, which is obtained as a special case. The confidence regions are constructed by inverting a continuous extension of the score test statistic standardized by expected information, which we show exists at points of singular information under regularity conditions. Similar results have previously only been obtained for scalar parameters, under conditions stronger than ours, and applications to mixed models have not been considered. In simulations our confidence regions have near-nominal coverage with as few as $n = 20$ independent observations, regardless of how close to the boundary the true parameter is. It is a corollary of our main results that the proposed test statistic has an asymptotic chi-square distribution with degrees of freedom equal to the number of tested parameters, even if they are on the boundary of the parameter set.