Abstract
Suppose X n is an observation, or average of observations, on a discretized signal ξ n that is measured at n time points. The random vector X n has a N(ξ n , σ2 n I) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator X n of ξ n can be improved by variable-selection. Consider the candidate estimator ξ n (A) whose i-th component equals the i-th component of X n whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A C p -estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a C p -estimator, have correct asymptotic coverage probabiligy for ξ n , and are geometrically smaller than or equal to the competing confidence balls centered at X n . The asymptotics are locally uniform in the parameters (ξ n , σ2 n ). The results illustrate an approach to inference after variable-selection.
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Beran, R. Confidence sets centered at C p -estimators. Ann Inst Stat Math 48, 1–15 (1996). https://doi.org/10.1007/BF00049285
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DOI: https://doi.org/10.1007/BF00049285