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Abstract:
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Estimation of tail probability is of interest in various applications. Given a parametric model, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, the large sample property is often not trustworthy for estimating small tail probabilities. We introduce a new estimator for the parameters, the Maximum Lq-Likelihood Estimator (MLqE), based on Havrda and Charvát entropy function, and apply it for estimating tail probabilities. Its behavior is characterized by the degree of distortion, q, applied to the assumed model; when q is close to 1 the new estimator approaches the usual MLE. We derive asymptotic properties of the MLqE and assess its efficiency, showing that it successfully trades bias for variance when the amount of information available is not large relative to the size of the tail probability to be estimated.
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