Observed information in semi-parametric models

In many semi-parametric models, 'regular' parameters can be estimated by (semi-parametric) maximum likelihood estimators. The asymptotic theory for such estimators has been developed for a number of models of practical interest, and is similar to the asymptotic theory for maximum likelihood estimators in classical parametric models. In particular, the maximum likelihood estimators are asymptotically normal, where the inverse of the 'efficient Fisher information matrix' gives the asymptotic covariance matrix. The latter matrix is the Fisher information matrix corrected for the presence of an infinite-dimensional nuisance parameter. See, for example, Bickel et al. (1993) for an extensive review of information bounds. See Gill (1989), Chang (1990), Gu and Zhang (1993), Qin (1993), van der Laan (1993), Qin and Lawless (1994), van der Vaart (1994a; 1994b; 1994c; 1996), Murphy (1995), Gill et al. (1995), Huang (1996), Parner (1998), Qin and Wong (1996) and Mammen and van de Geer (1997) for results on the asymptotics of particular maximum likelihood estimators. It is natural to use the asymptotic normality of the estimator in order to form confidence intervals and test statistics. This requires an estimator of the standard error or equivalently of the Fisher information matrix. In some specific cases the efficient Fisher information matrix is of closed form. For example, under the assumption that the observation time is independent of the covariates, Huang (1996) gives an explicit estimator of the asymptotic variance in a proportional hazards model applied to current status data. Sometimes the 'efficient score' or 'efficient influence function' is explicit. Then since the efficient Fisher information matrix is the covariance of the efficient score function, one may estimate the

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