Generalization of L-moments

The L-moments and trimmed L-moments (TL-moments) are both linear functions of order statistics. Exact variances and covariances expressions have derived for exact variances and covariances of sample L-moments and of sample TL-moments for any sample size in terms of first and second-order moments of order statistics from small conceptual sample sizes, which do not depend on the actual sample size. Moreover, we have established a theorem which characterises the normal distribution in terms of these second-order moments and the characterisation suggests a new test of normality. A method of estimation is derived by giving zero weights to extreme observations, this method called trimmed L-moments (TL-moments). TL-moments have certain advantages over L-moments and method of moments. They exist whether or not the mean exists and they are more robust to the presence of outliers. The Tukey symmetric lambda distribution is studied and the exponentially weighted moving average (EWMA) control charts are proposed to monitor the process mean and dispersion using the sample L-mean and sample L-scale and charts based on trimmed versions of the same statistics.