CONFIDENCE SETS CENTERED AT C_{p}-ESTIMATORS

RUDOLF BERAN

Department of Statistics, University of California, Berkeley, CA 94720-3860, U.S.A.

(Received March 18, 1994; revised July 17, 1995)

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}, σ_{n}^{2}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 ξ̂_{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 probability 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}, σ_{n}^{2}). The results illustrate an approach to inference after variable-selection.

Key words and phrases: Variable-selection, coverage probability, geometrical loss, locally uniform asymptotics.