On the informative value of the largest sample element of log-Gumbel distribution
Abstract
Extremes of stream flow and precipitation are commonly modeled by heavytailed
distributions. While scrutinizing annual flow maxima or the peaks over
threshold, the largest sample elements are quite often suspected to be low quality
data, outliers or values corresponding to much longer return periods than the observation
period. Since the interest is primarily in the estimation of the right tail (in the
case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements
of a series constitutes a problem of special concern. This study investigated the sensitivity
problem using the log-Gumbel distribution by generating samples of different
sizes (n) and different values of the coefficient of variation by Monte Carlo experiments.
Parameters of the log-Gumbel distribution were estimated by the probability
weighted moments (PWMs) method, method of moments (MOMs) and
maximum likelihood method (MLM), both for complete samples and the samples
deprived of their largest elements. In the latter case, the distribution censored by the
non-exceedance probability threshold, FT , was considered. Using FT instead of the
censored threshold T creates possibility of controlling estimator property. The effect
of the FT value on the performance of the quantile estimates was then examined. It
is shown that right censoring of data need not reduce an accuracy of large quantile
estimates if the method of PWMs or MOMs is employed. Moreover allowing bias
of estimates one can get the gain in variance and in mean square error of large
quantiles even if ML method is used.
Subject
floodslog-Gumber distribution
estimation methods
bias
Monte Carlo simulation
truncation
censoring
non-exceedance probability threshold