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Biometrika 1999 86(4):881-897; doi:10.1093/biomet/86.4.881
© 1999 by Biometrika Trust
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High-order accurate methods for retrospective sampling problems

S WangA1 and RJ CarrollA2

Department of Statistics, Texas A&M University, College Station, TX 77843-3143, USA A1 E-mail: sjwang@stat.tamu.edu A2 E-mail: carroll@stat.tamu.edu

In this paper we discuss the relationship between prospective and retrospective sampling problems. Estimates of the parameter of interest may be obtained by solving suitable estimating equations under both sampling schemes. Most common examples of such estimates include the maximum likelihood estimates. Some classical results and more recent development of the first-order asymptotic relationship between the estimators are reviewed. High-order expansions are given for the distributions of the retrospective estimators. Expansions for the marginal distributions of interest are obtained for both prospective and retrospective data. Furthermore, it is shown that the two expansions are asymptotically equal, at least up to order O(n-1). This implies that readily available prospective saddlepoint methods may be applied to the analysis of retrospective data without loss of high-order accuracy. The results are also briefly illustrated numerically.

Key Words: case-control sampling; choice-based sampling; logistic regression; prospective sampling; saddlepoint approximation; Taylor expansion


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