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

Paul Kabaila; Khageswor Giri

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arXiv · 1111.2113

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

Paul Kabaila·Khageswor Giri

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Abstract

Consider a linear regression model with n-dimensional response vector, regression parameter β= (β_1, ..., β_p) and independent and identically N(0, σ^2) distributed errors. Suppose that the parameter of interest is θ= a^T βwhere a is a specified vector. Define the parameter τ= c^T β- t where c and t are specified. Also suppose that we have uncertain prior information that τ= 0. Part of our evaluation of a frequentist confidence interval for θis the ratio (expected length of this confidence interval)/(expected length of standard 1-αconfidence interval), which we call the scaled expected length of this interval. We say that a 1-αconfidence interval for θutilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when τ= 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-αconfidence 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 \hatβdenote the least squares estimator of β. Also let \hatΘ= a^T \hatβand \hatτ= c^T \hatβ- t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(\hatΘ, \hatτ)| increases and n-p decreases.

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Paul Kabaila, Khageswor Giri. 2012-04-02. Further properties of frequentist confidence intervals in regression that utilize uncertain prior information. https://arxiv.org/abs/1111.2113

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