A method for simulating Burr type III and type XII distributions through \(L\)-moments and \(L\)-correlations (Q469805)

Summary: This paper derives the Burr Type III and Type XII family of distributions in the contexts of univariate \(L\)-moments and the \(L\)-correlations. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of \(L\)-skew, \(L\)-kurtosis, and \(L\)-correlations. The procedure can be applied in a variety of settings such as statistical modeling (e.g., forestry, fracture roughness, life testing, operational risk, etc.) and Monte Carlo or simulation studies. Numerical examples are provided to demonstrate that \(L\)-moment-based Burr distributions are superior to their conventional moment-based analogs in terms of estimation and distribution fitting. Evaluation of the proposed procedure also demonstrates that the estimates of \(L\)-skew, \(L\)-kurtosis, and \(L\)-correlation are substantially superior to their conventional product moment-based counterparts of skew, kurtosis, and Pearson correlations in terms of relative bias and relative efficiency -- most notably when heavy-tailed distributions are of concern.