Simple Counterexample to the Bootstrap

The bootstrap of the maximum likelihood estimator of the mean of a sample of iid normal random variables with mean µ and variance one is not asymptotically correct to first order when the mean is restricted to be nonnegative. The problem occurs when the true value of the mean µ equals zero. This counterexample to the bootstrap generalizes to a wide variety of estimation problems in which the true parameter may be on the boundary of the parameter space. We provide some alternatives to the bootstrap that are asymptotically correct to first order.

We consider two types of bootstrap percentile confidence intervals in the above example. We find that they both have asymptotic coverage probability that exceeds the nominal asymptotic level when the true value of the mean it equals zero.