L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics

Abstract

SUMMARY L‐moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis described in Section 2, and new methods of parameter estimation for several distributions. The theory of L‐moments parallels the theory of (conventional) moments, as this list of applications might suggest. The main advantage of L‐moments over conventional moments is that L‐moments, being linear functions of the data, suffer less from the effects of sampling variability: L‐moments are more robust than conventional moments to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. L‐moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates.

Present-day and future mediterranean precipitation extremes assessed by different statistical approaches

“…However, the location parameter can also be implicitly incorporated into the scale parameter (Coles 2001). We estimate the GPD parameters from the sample of observed or simulated precipitation extremes (exceeding a certain threshold which is defined further below) for the present-day or future reference period applying the method of L-moments which is relatively robust in the light of potential outliers and small rainfall samples over the dry Mediterranean area (Hosking 1990;Paeth and Hense 2005). Then, the RVs of daily extremes at various return times (RTs) are determined as (1-1/RT) quantiles of the cumulative GPD function taking into account the probability of occurrence, i.e.…”

“…In contrast to quantiles, this approach reduces sampling errors (Schönwiese 2006) and allows for the extrapolation towards return periods outside the sample period, however, with clearly increased domains of uncertainty. A drawback is that the parameters of the distribution have to be estimated from typically small samples (Hosking 1990). In this case, we fit the GPD to daily extremes exceeding a certain threshold.…”

Statistical Variability and Persistence Change in Daily Air Temperature Time Series from High Latitude Arctic Stations

“…For the normal distribution, L-skewness is equal to zero (as is the case for the classical skewness definition), while L-kurtosis is equal to 0.1226; higher L-moments of odd values can also be computed and regarded as generalized measures of distribution symmetry (HOSKING 1990). A wide range of applications as well as theoretical studies have proven L-moments to be particularly robust and accurate and to outperform other methods (HOSKING 2006;HOSKING and WALLIS 1997;LOUCKS and VAN BEEK 2005;AKRAM and HAYAT 2014).…”

Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

“…This is the standard half logistic distribution introduced and discussed in detail by Balakrishnan (1985); see also Balakrishnan and Cohen (1991). It should be mentioned here that Hosking (1986Hosking ( , 1990 proposed, discussed and applied many such generalizations of standard distributions.…”