Minimum L^q-distance estimators for non-normalized parametric models
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We propose and investigate a new estimation method for the parameters of models consisting of smooth density functions on the positive half axis. The procedure is based on a recently introduced characterization result for the respective probability distributions, and is to be classified as a minimum distance estimator, incorporating as a distance function the L^q-norm. Throughout, we deal rigorously with issues of existence and measurability of these implicitly defined estimators. Moreover, we provide consistency results in a common asymptotic setting, and compare our new method with classical estimators for the exponential-, the Rayleigh-, and the Burr Type XII distribution in Monte Carlo simulation studies. The procedure fares extraordinarily well in terms of the bias of the estimators and, in the case of the Burr distribution, where computational issues occur with the maximum likelihood estimator for very small sample sizes, the new method has no trouble in computations.
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