Computer-intensive Conditional Inference

Conditional inference is an intrinsic part of statistical theory, though not routinely of statistical practice. Conditioning has two principal objectives; (i) elimination of nuisance parameters, (ii) ensuring relevance of inference to observed sample data, through conditioning on an ancillary statistic, when such a statistic exists. Apart from formal difficulties with conditional inference, related to such issues as non-uniqueness of ancillary statistics, practical difficulties often arise, as calculating a conditional sampling distribution is typically not easy. Much interest therefore lies in inference procedures which are stable, that is, which are based on a statistic which has, to some high order in the sample size, the same repeated sampling behaviour unconditionally and conditional on the value of the appropriate conditioning statistic. Accurate approximation to an exact conditional inference can then be achieved by considering the marginal distribution of the stable statistic, ignoring the relevant conditioning. The principal approach to approximation of an intractable exact conditional inference by this route lies in developments in higher-order small-sample likelihood asymptotics. An alternative approach which we consider in this talk uses marginal simulation of the sampling distribution of an appropriately chosen stable statistic to mimic its conditional distribution. We offer theoretical results and empirical guidance to approximate conditioning by the computer-intensive route, for parametric inference on both scalar and vector interest parameters, in the presence of nuisance parameters. A key context where conditioning is used to eliminate nuisance parameters concerns inference in a multi-parameter exponential family setting. Here, computer-intensive methods yield third-order accuracy in approximation of exact conditional inference, under an appropriate handling of the nuisance parameter in the marginal simulation. In the ancillary statistic context, typically only second-order accuracy can be obtained via the marginal distribution of a stable statistic, though in practice excellent approximation is seen in many settings. We aim to provide general recommendations on the effectiveness of marginal simulation approaches to approximation of conditional inference. For one-sided inference on a scalar interest parameter, very effective approximations are obtained by simulating the marginal distribution of the signed root likelihood ratio statistic. With a vector interest parameter, good results are obtained by simulating the marginal expectation of the likelihood ratio statistic and making an empirical Bartlett correction. These approaches extend readily to more complex model settings, such as those involving high-dimensional parameters and where composite likelihood approaches are necessary