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Abstract:
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We propose a weighted maximum likelihood methodology for parametric inference and model selection based on a weighted and penalized Kullback-Leibler distance function. Weighted likelihood is an estimation method worked with by several authors, from different perspectives, including Hjort (1994b), Hjort and Jones (1996), Loader (1996), Eguchi and Copas (1998), Claeskens & Hjort (2008) and Schweder & Hjort (2016). In these approaches, the weight-function is predefined and independent of the actual data. New in our current approach is that this weight-function can also be data-driven, i.e. determined from the same data as being used to estimate the model parameters and perform model selection. In principle, any data-driven non-negative weight function can be used, for example to bring extra estimation efforts into certain regions of the sample space, or to down-weight more extreme values for robustness purposes, or a combination of these. Theoretical properties of the resulting estimator and accompanying model selection criterion are given, together with several illustrations of the proposed methodology for simulated and real case examples of i.i.d. and regression type of data.
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