Abstract
Many versions of weighted likelihood have been studied in the literature. The weighted likelihood that we are interested in was introduced to embrace formally a variety of statistical procedures that trade bias for precision. Unlike its classical counterpart, the weighted likelihood combines all relevant information while inheriting many of its desirable features including good asymptotic properties. However, in order to be effective, the weights involved in its construction need to be chosen adaptively. Wang and Zidek (2005) propose to choose adaptive likelihood weights by cross-validation. They have shown that the weighted likelihood estimator (WLE) with cross-validated weights is weakly consistent and asymptotically normal. In this paper, we prove the weak consistency and asymptotic normality of the adaptive WLE under much weaker assumptions. A simulation on the nonlinear case of the WLE is also provided.
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