Confidence Sets for a level set in linear regression

TL;DR

This paper develops a method for constructing confidence sets for level sets in linear regression, extending to other parametric models with monotonic link functions, and demonstrates its broad applicability and ease of use.

Contribution

It introduces a simple approach to create confidence sets for level sets in linear regression, applicable to various parametric models with monotonic link functions.

Findings

Confidence sets can be derived from simultaneous confidence bands.

Method is easily applicable to generalized linear and mixed models.

Examples illustrate the method's practical utility.

Abstract

Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It is realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this paper, the construction of confidence sets for the level set of linear regression is considered. In particular, $1-\alpha$ level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding $1-\alpha$ level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore the method proposed in this paper is widely applicable. Examples are given to illustrate the method.