| Abstract: | We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ0 in the presence of a (possibly infinite dimensional) nuisance parameter ψ0. It is supposed that the value taken by θ0 does not restrict the value that ψ0 may take and vice-versa. Given a sensible estimate ψn of ψ0, profile bootstrap confidence interval for θ0 is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ0=ψn. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ0 and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity. |
