L-moments: Analysis and estimation of distributions using linear combinations of order statistics.

*(English)*Zbl 0703.62018Summary: 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.

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.

##### MSC:

62E15 | Exact distribution theory in statistics |

62G05 | Nonparametric estimation |

62G30 | Order statistics; empirical distribution functions |