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.
Pant, Mohan D. and Headrick, Todd C. "A Method for Simulating Burr Type III and Type XII Distributions through L-Moments and L-Correlations." (Jan 2013).