Valid Confidence Intervals In Regression After Variable Selection
We consider a linear regression model with regression parameters ( 1,..., p) and error variance parameter 2. Our aim is to find a confidence interval with minimum coverage probability 1 for a parameter of interest 1 in the presence of nuisance parameters ( 2,..., p, 2). We consider two confidence intervals, the first of which is the standard confidence interval for 1 with coverage probability 1 . The second confidence interval for 1 is obtained after a variable selection procedure has been applied to p. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 . The confidence intervals are compared using a risk function that is defined as a scaled version of the expected length of the confidence interval. We show that, subject to certain conditions including that (dimension of response vector) p is small, the second confidence interval is preferable to the first when we anticipate (without being certain) that p / is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.
Volume (Year): 14 (1998)
Issue (Month): 04 (August)
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