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

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

2011-11-09

arxiv.org/abs/1111.2113v3

Statistics

Methodology

The exposition has been improved

Scientific paper

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.

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