Confidence sets centered at C-p-estimators

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(n)(2)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)over cap>(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 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.