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Full-Text Articles in Statistics and Probability
Bayesian Hierarchical Modeling With 3pno Item Response Models, Yanyan Sheng, Todd Christopher Headrick
Bayesian Hierarchical Modeling With 3pno Item Response Models, Yanyan Sheng, Todd Christopher Headrick
Todd Christopher Headrick
Fully Bayesian estimation has been developed for unidimensional IRT models. In this context, prior distributions can be specified in a hierarchical manner so that item hyperparameters are unknown and yet still have their own priors. This type of hierarchical modeling is useful in terms of the three-parameter IRT model as it reduces the difficulty of specifying model hyperparameters that lead to adequate prior distributions. Further, hierarchical modeling ameliorates the noncovergence problem associated with nonhierarchical models when appropriate prior information is not available. As such, a Fortran subroutine is provided to implement a hierarchical modeling procedure associated with the three-parameter normal …
An L-Moment Based Characterization Of The Family Of Dagum Distributions, Mohan D. Pant, Todd C. Headrick
An L-Moment Based Characterization Of The Family Of Dagum Distributions, Mohan D. Pant, Todd C. Headrick
Todd Christopher Headrick
This paper introduces a method for simulating univariate and multivariate Dagum distributions through the method of 𝐿-moments and 𝐿-correlations. 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 𝐿-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, …
Simulating Multivariate G-And-H Distributions, Rhonda K. Kowalchuk, Todd C. Headrick
Simulating Multivariate G-And-H Distributions, Rhonda K. Kowalchuk, Todd C. Headrick
Todd Christopher Headrick
The Tukey family of g-and-h distributions is often used to model univariate real-world data. There is a paucity of research demonstrating appropriate multivariate data generation using the g-and-h family of distributions with specified correlations. Therefore, the methodology and algorithms are presented to extend the g-and-h family from univariate to multivariate data generation. An example is provided along with a Monte Carlo simulation demonstrating the methodology. In addition, algorithms written in Mathematica 7.0 are available from the authors for implementing the procedure.
Statistical Simulation: Power Method Polynomials And Other Transformations, Todd C. Headrick
Statistical Simulation: Power Method Polynomials And Other Transformations, Todd C. Headrick
Todd Christopher Headrick
Although power method polynomials based on the standard normal distributions have been used in many different contexts for the past 30 years, it was not until recently that the probability density function (pdf) and cumulative distribution function (cdf) were derived and made available. Focusing on both univariate and multivariate nonnormal data generation, Statistical Simulation: Power Method Polynomials and Other Transformations presents techniques for conducting a Monte Carlo simulation study. It shows how to use power method polynomials for simulating univariate and multivariate nonnormal distributions with specified cumulants and correlation matrices. The book first explores the methodology underlying the power method, …
A Method For Simulating Multivariate Non-Normal Distributions With Specified Standarized Cumulants And Intraclass Correlation Coefficients, Todd C. Headrick, Bruno D. Zumbo
A Method For Simulating Multivariate Non-Normal Distributions With Specified Standarized Cumulants And Intraclass Correlation Coefficients, Todd C. Headrick, Bruno D. Zumbo
Todd Christopher Headrick
Intraclass correlation coefficients (ICCs) are commonly used indices in subject areas such as biometrics, longitudinal data analysis, measurement theory, quality control, and survey research. The properties of the ICCs most often used are derived under the assumption of normality. However, real-world data often violate the normality assumption. In view of this, a computationally efficient procedure is developed for simulating multivariate non normal continuous distributions with specified (a) standardized cumulants, (b) Pearson intercorrelations, and (c) ICCs. The linear model specified is a two-factor design with either fixed or random effects. A numerical example is worked and the results of a Monte …
Simulating Controlled Variate And Rank Correlations Based On The Power Method Transformation, Todd C. Headrick, Simon Y. Aman, T. Mark Beasley
Simulating Controlled Variate And Rank Correlations Based On The Power Method Transformation, Todd C. Headrick, Simon Y. Aman, T. Mark Beasley
Todd Christopher Headrick
The power method transformation is a popular algorithm used for simulating correlated non normal continuous variates because of its simplicity and ease of execution. Statistical models may consist of continuous and (or) ranked variates. In view of this, the methodology is derived for simulating controlled correlation structures between non normal (a) variates, (b) ranks, and (c) variates with ranks in the context of the power method. The correlation structure between variate-values and their associated rank-order is also derived for the power method. As such, a measure of the potential loss of information is provided when ranks are used in place …
The Power Method Transformation: Its Probability Density Function, Distribution Function, And Its Further Use For Fitting Data, Todd C. Headrick, Rhonda K. Kowalchuk
The Power Method Transformation: Its Probability Density Function, Distribution Function, And Its Further Use For Fitting Data, Todd C. Headrick, Rhonda K. Kowalchuk
Todd Christopher Headrick
The power method polynomial transformation is a popular algorithm used for simulating non-normal distributions because of its simplicity and ease of execution. The primary limitations of the power method transformation are that its probability density function (pdf) and cumulative distribution function (cdf) are unknown. In view of this, the power method’s pdf and cdf are derived in general form. More specific properties are also derived for determining if a given transformation will also have an associated pdf in the context of polynomials of order three and five. Numerical examples and parametric plots of power method densities are provided to confirm …
An Investigation Of The Rank Transformation In Multple Regression, Todd C. Headrick, Ourania Rotou
An Investigation Of The Rank Transformation In Multple Regression, Todd C. Headrick, Ourania Rotou
Todd Christopher Headrick
Real world data often fail to meet the underlying assumptions of normal statistical theory. The rank transformation (RT) procedure is recommended and used in the context of multiple regression analysis when the assumption of normality is violated. There is no general supporting theory of the RT. In view of this, the current study examined the Type I error and power properties of the RT in terms of multiple regression. The investigation included both additive and nonadditive models. Results indicated that there were severely inflated Type I error rates associated with the RT procedure under both normal and nonnormal distributions (e.g., …
Simulating Correlated Multivariate Nonnormal Distributions: Extending The Fleishman Power Method, Todd C. Headrick, Shlomo S. Sawilowsky
Simulating Correlated Multivariate Nonnormal Distributions: Extending The Fleishman Power Method, Todd C. Headrick, Shlomo S. Sawilowsky
Todd Christopher Headrick
A procedure for generating multivariate nonnormal distributions is proposed. Our procedure generates average values of intercorrelations much closer to population parameters than competing procedures for skewed and/or heavy tailed distributions and for small sample sizes. Also, it eliminates the necessity of conducting a factorization procedure on the population correlation matrix that underlies the random deviates, and it is simpler to code in a programming language (e.g,, FORTRAN). Numerical examples demonstrating the procedures are given. Monte Carlo results indicate our procedure yields excellent agreement between population parameters and average values of intercorrelation, skew, and kurtosis.