Further properties of frequentist confidence intervals in regression that utilize uncertain prior information

TL;DR

This paper investigates properties of frequentist confidence intervals in linear regression that incorporate uncertain prior information, aiming to improve interval performance especially with high correlation and small sample sizes.

Contribution

It introduces new theoretical results and computational methods to evaluate and enhance confidence intervals utilizing uncertain prior information in regression models.

Findings

Performance improves with higher correlation between estimators.

Smaller sample sizes lead to better performance of the confidence intervals.

The method effectively reverts to standard intervals when prior information is contradicted.

Abstract

Consider a linear regression model with n-dimensional response vector, regression parameter \beta = (\beta_1, ..., \beta_p) and independent and identically N(0, \sigma^2) distributed errors. Suppose that the parameter of interest is \theta = a^T \beta where a is a specified vector. Define the parameter \tau = c^T \beta - t where c and t are specified. Also suppose that we have uncertain prior information that \tau = 0. Part of our evaluation of a frequentist confidence interval for \theta is the ratio (expected length of this confidence interval)/(expected length of standard 1-\alpha confidence interval), which we call the scaled expected length of this interval. We say that a 1-\alpha confidence interval for \theta utilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when \tau = 0, (b) the maximum value of the scaled expected length is not too much larger than 1 and (c) this confidence interval reverts to the standard 1-\alpha confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri, 2009, JSPI present a new method for finding such a confidence interval. Let \hat\beta denote the least squares estimator of \beta. Also let \hat\Theta = a^T \hat\beta and \hat\tau = c^T \hat\beta - t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(\hat\Theta, \hat\tau)| increases and n-p decreases.