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

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

Abstract.    Suppose Xn is an observation, or average of observations, on a discretized signal xin that is measured at n time points. The random vector Xn has a N(xin, sigma2n I) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator Xn of xin can be improved by variable-selection. Consider the candidate estimator ^xin(A) whose i-th component equals the i-th component of Xn whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A Cp-estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a Cp-estimator, have correct asymptotic coverage probability for xin, and are geometrically smaller than or equal to the competing confidence balls centered at Xn. The asymptotics are locally uniform in the parameters (xin, sigman2). The results illustrate an approach to inference after variable-selection.

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

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