On robustness of large quantile estimates of log-Gumbel and log-logistic distributions to largest element of the observation series: Monte Carlo results vs. first order approximation.

Abstract

Robustness of large quantile estimates to the largest element in a sample of methods of moments (MOM) and L-moments (LMM) was evaluated and compared. Quantiles were estimated by log-logistic and log-Gumbel distributions. Both are lower bounded and two-parameter distributions, with the coefficient of variation (CV) serving as the shape parameter. In addition, the results of these two methods were compared with those of the maximum likelihood method (MLM). Since identification and elimination of the outliers in a single sample require the knowledge of the sample’s parent distribution which is unknown, one estimates it by using the parameter estimation method which is relatively robust to the largest element in the sample. In practice this means that the method should be robust to extreme elements (including outliers) in a sample.

The effect of dropping the largest element of the series on the large quantile values was assessed for various “coefficient of variation (CV) / sample size (N)” combinations. To that end, Monte-Carlo sampling experiments were applied. The results were compared with those obtained from the single “representative” sample, (the first order approximation), i.e., consisting of both the average values (Ex _i ) for every position (i) of an ordered sample and the theoretical quantiles based on the plotting formula (PP).

The ML-estimates of large quantiles were found to be most robust to largest element of samples except for a small sample where MOM-estimates were more robust. Comparing the performance of two other methods in respect to the large quantiles estimation, MOM was found to be more robust for small and moderate samples drawn from distributions with zero lower bound as shown for log-Gumbel and log-logistic distributions. The results from “representative” samples were fairly compatible with the M-C simulation results. The Ex-sample results were closer to the M-C results for smaller CV-values, and to the PP-sample results for greater CV values.