Approximate Studentization with marginal and conditional inference

Many inference problems lead naturally to a marginal or conditional measure of departure that depends on a nuisance parameter. As a device for first‐order elimination of the nuisance parameter, we suggest averaging with respect to an exact or approximate confidence distribution function. It is shown that for many standard problems where an exact answer is available by other methods, the averaging method reproduces the exact answer. Moreover, for the gamma‐mean problem, where the exact answer is not explicitly available, the averaging method gives results that agree closely with those obtained from higher‐order asymptotic methods. Examples are discussed; detailed asymptotic calculations will be examined elsewhere.