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CONFIDENCE SETS CENTERED AT *Cp*-ESTIMATORS

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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 *xi*_{n} that is measured at
*n* time points. The random vector *X*_{n} has a *N*(*xi*_{n},
*sigma*^{2}_{n} *I*) distribution, the mean and variance being unknown.
Under squared error loss, the unbiased estimator *X*_{n} of *xi*_{n}
can be improved by variable-selection. Consider the candidate
estimator ^{^}*xi*_{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 *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 *xi*_{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 (*xi*_{n}, *sigma*_{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.

**Source**
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