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Biometrika 1996 83(1):1-13; doi:10.1093/biomet/83.1.1
© 1996 by Biometrika Trust
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Projected score methods for approximating conditional scores

RICHARD P. WATERMAN and BRUCE G. LINDSAY

Department of Statistics, University of Pennsylvania Philadelphia, Pennsylvania 19104, U.S.A.
Department of Statistics, Pennsylvania State University University Park, Pennsylvania 16802, U.S.A.

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.

Key Words: Bhattacharyya basis • Estimating equation • Exponential family • Information optimality • Nuisance parameter


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