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Empirical likelihood and general estimating equations. (English) Zbl 0799.62049

Summary: For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. A. Owen [ibid. 18, No. 1, 90-120 (1990; Zbl 0712.62040), and ibid. 19, No. 4, 1725-1747 (1991; see the preceding review Zbl 0799.62048)] showed that empirical likelihood ratio statistics for various parameters $\theta \left(F\right)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods.

Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.

MSC:
 62G20 Nonparametric asymptotic efficiency 62E20 Asymptotic distribution theory in statistics 62G05 Nonparametric estimation 62G10 Nonparametric hypothesis testing