On robustness of large quantile estimates to largest elements of the observation series
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The robustness of large quantile estimates of largest elements in a small sample by the methods of moments (MOM), L-moments (LMM) and maximum likelihood (MLM) was evaluated and compared. Bias (B) and mean square error (MSE) were used to measure the estimation methods performance. Quantiles were estimated by eight two-parameter probability distributions with the variation coefficient being the shape parameter. The effect of dropping largest elements of the series on large quantile values was assessed for various variation coefficient (C v ) /sample size (n) 'combinations' with n = 30 as the basic value. To that end, both the Monte Carlo sampling experiments and an asymprotic approach consisting in distribution truncation were applied. In general, both sampling and asymptotic approaches point to MLN as the most robust method of the three considered, with respect to bias of large quantiles. Comparing the performance of two other methods, the MOM estimates were found to be more robust for small and moderate hydrological samples drawn from distributions with zero lower-bound than were the LMM estimates. Extending the evaluation to outliers, it was shown that all the above findings remain valid. However, using the MSE variation as a measure of performance, the LMM was found to be the best for most distribution/variation coefficient combinations, whereas MOM was found to be the worst. Moreover, removal of the largest sample element need not result in a loss of estimation efficiency. The gain in accuracy is observed for the heavy-tailed and log-normal distributions, being particularly distinctive for LMM. In practice, while dealing with a single sample deprived of its largest element, one should choose the estimation method giving the lowest MSE of large quantiles. For n = 30 and several distribution/variation coefficient combinations, the MLM outperformed the two other methods in this respect and its supremacy grew with sample size, while MOM was usually the worst. Copyright 2006 John Wiley & Sons, Ltd.
