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Projected score methods for approximating conditional scores

SUMMARY This paper extends the projected score methods of Small & McLeish (1989). It is shown that the conditional score function may be approximated, with arbitrarily small stochastic error, in terms of a natural basis for the space of centred likelihood ratios. The utility of using this basis is established by identifying a U-statistic representation theorem and a class of expectation identities for the basis elements, making higher order asymptotics more tractable. The results are applied to a canonical exponential family model, where it is shown that the projected scores with estimated nuisance parameters can provide an accurate approximation to the conditional score function.