Mohan D. Pant

Characterizing Tukey ℎ and ℎℎ-Distributions through -Moments and the -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 Nonnormal Distributions with Specified L-Skew, L-Kurtosis, and L-Correlation

This paper introduces two families of distributions referred to as the symmetric κ and asymmetric κL-κR distributions. The families are based on transformations of standard logistic pseudo-random deviates. The primary focus of the theoretical development is in the contexts of L-moments and the L-correlation. Also included is the development of a method for specifying distributions with controlled degrees of L-skew, L-kurtosis, and L-correlation. The method can be applied in a variety of settings such as Monte Carlo studies, simulation, or modeling events. It is also demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional productmoment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when moderate-to-heavy-tailed distributions are of concern.

A Method for Simulating Burr Type III and Type XII Distributions through -Moments and -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 L-Moment-Based Analog for the Tukey g-h, g, h, and h-h 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 ,a ndh-h system of distributions. The system also consists of four classes of distributions and is referred to as � iasymmetric γ-κ, � iilog-logistic γ, � iiisymmetric κ ,a ndivasymmetric κL-κR. The system can be used in a variety of settings such as simulation or modeling events—most notably when heavy- tailed distributions are of interest. A procedure is also described for simulating γ-κ, γ, κ ,a ndκL-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 ,a ndκL-κR distributions are substantially superior to their corresponding conventional product- moment estimators in terms of relative bias and relative standard error.

On the Order Statistics of Standard Normal-Based Power Method Distributions

This paper derives a procedure for determining the expectations of order statistics associated with the standard normal distribution (𝑍) and its powers of order three and five (𝑍3 and 𝑍5). The procedure is demonstrated for sample sizes of 𝑛≤9. It is shown that 𝑍3 and 𝑍5 have expectations of order statistics that are functions of the expectations for 𝑍 and can be expressed in terms of explicit elementary functions for sample sizes of 𝑛≤5. For sample sizes of 𝑛=6,7 the expectations of the order statistics for 𝑍, 𝑍3, and 𝑍5 only require a single remainder term.

A Doubling Method for the Generalized Lambda Distribution

This paper introduces a new family of generalized lambda distributions (GLDs) based on a method of doubling symmetric GLDs. The focus of the development is in the context of L-moments and L-correlation theory. As such, included is the development of a procedure for specifying double GLDs 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 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.

An L-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.

A Characterization of the Burr Type III and Type XII Distributions through the Method of Percentiles and the Spearman Correlation

ABSTRACT A characterization of Burr Type III and Type XII distributions based on the method of percentiles (MOP) is introduced and contrasted with the method of (conventional) moments (MOM) in the context of estimation and fitting theoretical and empirical distributions. The methodology is based on simulating the Burr Type III and Type XII distributions with specified values of medians, inter-decile ranges, left-right tail-weight ratios, tail-weight factors, and Spearman correlations. Simulation results demonstrate that the MOP-based Burr Type III and Type XII distributions are substantially superior to their (conventional) MOM-based counterparts in terms of relative bias and relative efficiency.

Using a Generalized Linear Mixed Model Approach to Explore the Role of Age, Motor Proficiency, and Cognitive Styles in Children's Reach Estimation Accuracy

The purpose was to use a multi-level statistical technique to analyze how childrens age, motor proficiency, and cognitive styles interact to affect accuracy on reach estimation tasks via Motor Imagery and Visual Imagery. Results from the Generalized Linear Mixed Model analysis (GLMM) indicated that only the 7-year-old age group had significant random intercepts for both tasks. Motor proficiency predicted accuracy in reach tasks, and cognitive styles (object scale) predicted accuracy in the motor imagery task. GLMM analysis is suitable to explore age and other parameters of development. In this case, it allowed an assessment of motor proficiency interacting with age to shape how children represent, plan, and act on the environment.

Characterizing Log-Logistic (LL) Distributions through Methods of Percentiles and L-Moments

The main purpose of this paper is to characterize the log-logistic (LL) distributions through the methods of percentiles and L-moments and contrast with the method of (product) moments. The method of (product) moments (MoM) has certain limitations when compared with method of percentiles (MoP) and method of L-moments (MoLM) in the context of fitting empirical and theoretical distributions and estimation of parameters, especially when distributions with greater departure from normality are involved. Systems of equations based on MoP and MoLM are derived. A methodology to simulate univariate LL distributions based on each of the two methods (MoP and MoLM) is developed and contrasted with MoM in terms of fitting distributions and estimation of parameters. Monte Carlo simulation results indicate that the MoPand MoLM-based LL distributions are superior to their MoM based counterparts in the context of fitting distributions and estimation of parameters. Mathematics Subject Classification: 62G30, 62H12, 62H20, 65C05, 65C10, 65C60, 78M05