A Doubling Technique for the Power Method Transformations

Power method polynomials are used for simulating non-normal distributions with specified product moments or L-moments. The power method is capable of producing distributions with extreme values of skew (L-skew) and kurtosis (L-kurtosis). However, these distributions can be extremely peaked and thus not representative of real-world data. To obviate this problem, two families of distributions are introduced based on a doubling technique with symmetric standard normal and logistic power method distributions. The primary focus of the methodology is in the context of L-moment theory. As such, L-moment based systems of equations are derived for simulating univariate and multivariate non-normal distributions with specified values of L-skew, L-kurtosis, and L-correlation. Evaluation of the proposed doubling technique indicates that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional product-moments in terms of relative bias and relative efficiency when extreme non-normal distributions are of concern

An L-Moment Based Characterization of the Family of Dagum Distributions

This paper introduces a method for simulating univariate and multivariate Dagum distributions through the method of L-moments and L-correlation. A method is developed for characterizing non-normal Dagum distributions with controlled degrees of L-skew, L-kurtosis, and L-correlations. The procedure can be applied in a variety of contexts such as statistical modeling (e.g., income distribution, personal wealth distributions, etc.) and Monte Carlo or simulation studies. Numerical examples are provided to demonstrate that -moment-based Dagum distributions are superior to their conventional moment-based analogs in terms of estimation and distribution fitting. Evaluation of the proposed method 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 correlation in terms of relative bias and relative efficiency–most notably in the context of heavy-tailed distributions

Uniform-based and triangular-based third-order power method distributions using a doubling technique distributions

Power method (PM) polynomials have been used for simulating non-normal distributions in a variety of settings such as toxicology research, price risk, business-cycle features, microarray analysis, computer adaptive testing, and structural equation modeling. A majority of the applications associated with the PM polynomials are based on the method of matching conventional moments (e.g., skew and kurtosis). However, estimators of skew and kurtosis can be (a) substantially biased, (b) highly dispersed, or (c) influenced by outliers. To address this limitation, two families of third-order PM distributions are developed through the method of -moments (Hosking, 1990) using a doubling technique (Morgenthaler & Tukey, 2000) and contrasted with the method of moments in the contexts of estimation of parameters. The methodology is based on simulating uniform- and triangular-based third-order PM distributions with specified values of -skew and - kurtosis. Monte Carlo simulation results indicate that the estimators based on method of Lmoments are superior to their conventional moment-based counterparts

Characterizing Tukey \u3ci\u3eh\u3c/i\u3e and \u3ci\u3ehh\u3c/i\u3e-Distributions through \u3ci\u3eL\u3c/i\u3e-Moments and the \u3ci\u3eL\u3c/i\u3e-Correlation

This paper introduces the Tukey family of symmetric h and asymmetric hh-distributions in the contexts of univariate L-moments and the L-correlation. 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 modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern

A Method for Simulating Burr Type III and Type XII Distributions through \u3ci\u3eL\u3c/i\u3e-Moments and \u3ci\u3eL\u3c/i\u3e-Correlations

This paper derives the Burr Type III and Type XII family of distributions in the contexts of univariate -moments and the - correlations. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of -skew, -kurtosis, and -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 -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 -skew, -kurtosis, and -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

A Logistic \u3ci\u3eL\u3c/i\u3e-Moment-Based Analog for the Tukey \u3ci\u3eg-h, g, h\u3c/i\u3e, and \u3ci\u3eh-h\u3c/i\u3e System of Distributions

This paper introduces a standard logistic L-moment-based system of distributions. The proposed system is an analog to the standard normal conventional moment-based Tukey g-h, g, h, and h-h system of distributions. The system also consists of four classes of distributions and is referred to as (i) asymmetric γ-κ, (ii) log-logistic γ, (iii) symmetric κ, and (iv) asymmetric κL-κR. The system can be used in a variety of settings such as simulation or modeling events—most notably when heavytailed distributions are of interest. A procedure is also described for simulating γ-κ, γ, κ, and κL-κR distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that estimates of L-skew, L-kurtosis, and L-correlation associated with the γ-κ, γ, κ, and κL-κR distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias and relative standard error

An \u3ci\u3eL\u3c/i\u3e-Moment-Based Analog for the Schmeiser-Deutsch Class of Distributions

This paper characterizes the conventional moment-based Schmeiser-Deutsch (S-D) class of distributions through the method of L-moments. The system can be used in a variety of settings such as simulation or modeling various processes. A procedure is also described for simulating S-D distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that the estimates of L-skew, L-kurtosis, and L-correlation associated with the S-D class of distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias—most notably when sample sizes are small