Goodness-of-fit techniques
Beta kernel quantile estimators of heavy-tailed loss distributions
Statistics and Computing
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In this paper we assess accuracy of some commonly used estimators of upper quantiles of a right skewed distribution under both parameter and model uncertainty. In particular, for each of log-normal, log-logistic, and log-double exponential distributions, we study the bias and mean squared error of the maximum likelihood estimator (MLE) of the upper quantiles under both the correct and incorrect model specifications. We also consider two data dependent or adaptive estimators. The first (tail-exponential) is based on fitting an exponential distribution to the highest 10-20 percent of the data. The second selects the best fitting likelihood-based model and uses the MLE obtained from that model. The simulation results provide some practical guidance concerning the estimation of the upper quantiles when one is uncertain about the underlying model. We found that the consequences of assuming log-normality when the true distribution is log-logistic or log-double exponential are not severe in moderate sample sizes. For extreme quantiles, no estimator was reliable in small samples. For large sample sizes the selection estimator performs fairly well. For small sample sizes the tail-exponential method is a good alternative. Presenting it and the MLE for the log-normal enables one to assess the potential effects of model uncertainty.