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Adaptive likelihood weights and mixtures of empirical distributions Plante, Jean-François

Abstract

Suppose that you must make inference about a population, but that data from m -1 similar populations are available. The weighted likelihood uses exponential weights to include all the available information into the inference. The contribution of each datum is discounted based on its dissimilarity with the target distribution. One could hope to elicitate the likelihood weights from scientific information, but using data-based weights is more pragmatic. To this day, no entirely satisfactory method has been found for determining likelihood weights from the data. We propose a way to determine the likelihood weights based on data. The suggested "MAMSE" weights are nonparametric and can be used as likelihood weights, or as mixing probabilities to define a mixture of empirical distributions. In both cases, using the MAMSE weights allows strength to be borrowed from the m -1 similar populations whose distribution may differ from the target. The MAMSE weights are defined for different types of data: univariate, censored and multivariate. In addition to their role for the likelihood, the MAMSE weights are used to define a weighted Kaplan-Meier estimate of the survival distribution and weighted co-efficients of correlation based on ranks. The maximum weighted pseudo-likelihood, a new method to fit a family of copulas, is also proposed. All these examples of inference using the MAMSE weights are shown to be asymptotically unbiased. Furthermore, simulations show that inference based on MAMSE-weighted methods can perform better than their unweighted counterparts. Hence, the adaptive weights we propose successfully trade bias for precision.

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