Browsing by Subject "robustness"
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Item Effect Size Matters: Empirical Investigations to Help Researchers Make Informed Decisions on Commonly Used Statistical Techniques(2011-02-22) Skidmore, Susana TroncosoThe present journal article formatted dissertation assessed the characteristics of effect sizes of commonly used statistical techniques. In the first study, the author examined the American Educational Research Journal (AERJ) and select American Psychological Association (APA) and American Counseling Association (ACA) journals to provide an historical account and synthesis of which statistical techniques were most prevalent in the fields of education and psychology. These reviews represented a total of 17,698 techniques recorded from 12,012 articles. Findings point to a general decrease in the use of the tvtest and ANOVA/ANCOVA and a general increase in the use of regression and factor/cluster analysis. In the second study, the author compared the efficacy of one Pearson r2 and seven multiple R2 correction formulas for the Pearson r2. The author computed adjustment bias and precision under 108 conditions (6 population p2 values, 3 shape conditions and 6 sample size conditions). The Pratt and the Olkin-Pratt Extended formulas more consistently provided unbiased estimates across sample sizes, p2 values and the shape conditions investigated. In the third study, the author evaluated the robustness of estimates of practical significance (n2, e2 and w2) in one-way between subjects univariate ANOVA. There were 360 simulation conditions (5 population Cohen's d values, 4 group proportion ratios, 3 shape conditions, 3 variance conditions, and 2 total sample size conditions) for each of three group configurations (2, 3 and 4 groups). Three indices of practical significance (n2, e2, w2) and two indices of statistical significance (Type I error and power) were computed for each of the 5,400, 000 (5,000 replications x 360 simulation conditions x 3 group configurations). Simulation findings for n2 under heterogeneous variance conditions indicated that for the k=2 and k=3 condition Cohen's d values up to 0.2 (up to 0.5 for k=4) tend to produce overestimated population n2 values. Under heterogeneous variance conditions for e2 and w2 at Cohen's d = 0.0 and 0.2, the negative variance pairing overestimated and the positive variance pairing underestimated the parameter n2 but at Cohen's d greater than or equal to 0.5, both the positive and negative variance conditions resulted in underestimated parameter n2 values.Item Guaranteed Verification of Finite Element Solutions of Heat Conduction(2012-07-16) Wang, DelinThis dissertation addresses the accuracy of a-posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy. The investigation shows that the main culprit behind the lack of robustness of residual estimators is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution. With the introduction of an elliptic reconstruction procedure, a new error estimator based on the solution of the elliptic reconstruction problem is invented to estimate the exact error measured in space-time C-norm for both semi-discrete and fully discrete finite element solutions to linear parabolic problem. For a fully discrete solution, a temporal error estimator is also introduced to evaluate the discretization error in the temporal field. In the meantime, the implicit Neumann subdomain residual estimator for elliptic equations, which involves the solution of the local residual problem, is combined with the elliptic reconstruction procedure to carry out a posteriori error estimation for the linear parabolic problem. Numerical examples are presented to illustrate the superconvergence properties in the elliptic reconstruction and the performance of the bounds based on the space-time C-norm. The results show that in the case of L^2 norm for smooth solution there is no superconvergence in elliptic reconstruction for linear element, and for singular solution the superconvergence does not exist for element of any order while in the case of energy norm the superconvergence always exists in elliptic reconstruction. The research also shows that the performance of the bounds based on space-time C-norm is robust, and in the case of fully discrete finite element solution the bounds for the temporal error are sharp.Item Multivariate Skew-t Distributions in Econometrics and Environmetrics(2012-02-14) Marchenko, Yulia V.This dissertation is composed of three articles describing novel approaches for analysis and modeling using multivariate skew-normal and skew-t distributions in econometrics and environmetrics. In the first article we introduce the Heckman selection-t model. Sample selection arises often as a result of the partial observability of the outcome of interest in a study. In the presence of sample selection, the observed data do not represent a random sample from the population, even after controlling for explanatory variables. Heckman introduced a sample-selection model to analyze such data and proposed a full maximum likelihood estimation method under the assumption of normality. The method was criticized in the literature because of its sensitivity to the normality assumption. In practice, data, such as income or expenditure data, often violate the normality assumption because of heavier tails. We first establish a new link between sample-selection models and recently studied families of extended skew-elliptical distributions. This then allows us to introduce a selection-t model, which models the error distribution using a Student?s t distribution. We study its properties and investigate the finite-sample performance of the maximum likelihood estimators for this model. We compare the performance of the selection-t model to the Heckman selection model and apply it to analyze ambulatory expenditures. In the second article we introduce a family of multivariate log-skew-elliptical distributions, extending the list of multivariate distributions with positive support. We investigate their probabilistic properties such as stochastic representations, marginal and conditional distributions, and existence of moments, as well as inferential properties. We demonstrate, for example, that as for the log-t distribution, the positive moments of the log-skew-t distribution do not exist. Our emphasis is on two special cases, the log-skew-normal and log-skew-t distributions, which we use to analyze U.S. precipitation data. Many commonly used statistical methods assume that data are normally distributed. This assumption is often violated in practice which prompted the development of more flexible distributions. In the third article we describe two such multivariate distributions, the skew-normal and the skew-t, and present commands for fitting univariate and multivariate skew-normal and skew-t regressions in the statistical software package Stata.