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, called Maximum L_q-Likelihood Estimator (ML_qE), based on Havrda and Charvát’s entropy function (Havrda and Charvát, 1967), and apply it for estimating tail probabilities. The ML_qE can be regarded as an extension of the traditional log-likelihood maximization procedure. Specifically, 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 new estimator and assess its efficiency, showing that the ML_qE successfully trades bias for precision when the amount of information available is not large relative to the size of the tail probability to be estimated. In fact, in a suitable framework, we show that when the distortion parameter q is properly chosen according to the sample size, the ratio of the Mean Squared Error of ML_qE over that of MLE converges to 0.  Monte Carlo simulations are carried out for the case of the exponential distribution, and the results confirm the theoretical finding.