A NOVEL ESTIMATION METHOD BASED ON MAXIMUM LIKELIHOOD
AdvisorKozubowski, Tomasz J.
Mathematics and Statistics
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The method of maximum likelihood (ML) is perhaps the most widely used statistical approach to estimate unknown parameters in a parametric setting. However, the required optimization of the likelihood function is rarely possible explicitly, and finding the estimators may be computationally challenging. On the other hand, maximum likelihood estimators are often simple to compute when the sample size is equal to one. Based on this observation, we propose a novel approach to estimation, where each individual observation in a random sample is used to derive an estimator of an unknown parameter using the ML principle. These individual estimators are then put together as a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood function evaluated at the estimators based on each observation. It turns out that this method can be related to a Bayesian approach, where the prior distribution is data driven. In case of estimating a location parameter of a unimodal density, the prior distribution is the empirical distribution of the sample, and converges to the true distribution that generated the data as the sample size increases. We provide several examples illustrating the new method, and conduct simulation studies to assess the performance of the estimators. It turns out that this straightforward methodology produces consistent estimators, which seem to be comparable with those obtained by the ML method in large sample setting, and may actually outperform the latter when the sample size is small.