Confidence sets centered at C p -estimators

Suppose X_n is an observation, or average of observations, on a discretized signal _n that is measured at n time points. The random vector X_n has a N(_n, ²_nI) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator X_n of _n can be improved by variable-selection. Consider the candidate estimator _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 _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 (_n, ²_n). The results illustrate an approach to inference after variable-selection.