The Theory of Probability Weighted Moments

Probability weighted moments (PWMs) are expectations of certain functions of a random variable. They can be defined for any random variable whose mean exists and were
devised by Greenwood et al. (1979) primarily as an aid to estimating the parameters of the Wakeby distribution. Greenwood et al.'s use of PWMs is, however. but one application of
a general theory which is founded on PWMs and covers:

• the summarization and description of theoretical probability distributions:
• the summarization and description of observed data samples:
• nonparametric estimation of the underlying distribution of an observed sample:
• estimation of parameters and quantifies of probability distributions:
• hypothesis tests for probability distributions.

The theory involves such established and efficient procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the new measures of skewness and kurtosis described in section 3, and new methods of parameter estimation for several distributions. The theory of PWMs parallels the theory of (conventional) moments. as the above list of applications might suggest. The main advantage of PWMs over conventional moments is that PWMs, being linear functions of the data, suffer less from the effects of sampling variability: PWMs are more robust than conventional moments to outliers in the data, enable more secure inferences to he made from small samples about an underlying probability distribution, and frequently yield more efficient parameter estimates than the conventional moment estimates.