It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, are asymptotically normal with means equal to the true parameter values (although strictly speaking appropriate regularity conditions need to be verified for the asymptotic normality to hold, which I can't find a reference formally showing this). Further still, this convergence to normality is convergence in distribution, which on its own does not imply that the biases of the MLEs vanish. For the asymptotic normality to imply convergence of the means (not in the mean) the MLEs have to be uniformly integrable (may be uniform integrability can be relaxed). There are papers dealing with the biases, but has anyone actually verified that the MLEs are, in fact, asymptotically unbiased (in the sense that $E[\hat{\theta}_n]\to\theta$ as $n\to\infty$)?

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    $\begingroup$ Since asymptotic unbiasedness is proved using more general properties of the MLE, what would "verify" mean here? Simulation to that effect? Some "direct" calculation of the limit? $\endgroup$ Nov 26, 2014 at 19:38
  • $\begingroup$ My question is whether it is actually true that the two-parameter Weibull MLEs are asymptotically unbiased? Is there a reference where this is shown to be true rigorously? $\endgroup$
    – Jeff
    Nov 27, 2014 at 20:16


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