A deviance-based criterion for model selection in GLM

Model selection is the most persuasive problem in generalized linear models. A model selection criterion based on deviance called the deviance-based criterion (DBC) is proposed. The DBC is obtained by penalizing the difference between the deviance of the fitted model and the full model. Under certain weak conditions, DBC is shown to be a consistent model selection criterion in the sense that with probability approaching to one, the selected model asymptotically equals the optimal model relating response and predictors. Further, the use of DBC in link function selection is also discussed. We compare the proposed model selection criterion with existing methods. The small sample efficiency of proposed model selection criterion is evaluated by the simulation study.     

Multivariate control chart based on the highest possibility region

The T ² control chart is widely adopted in multivariate statistical process control. However, when dealing with asymmetrical or multimodal distributions using the traditional T ² control chart, some points with relatively high occurrence possibility might be excluded, while some points with relatively low occurrence possibility might be accepted. Motived by the thought of the highest posterior density credible region, we develop a control chart based on the highest possibility region to solve this problem. It is shown that the proposed multivariate control chart will not only meet the false alarm requirement, but also ensure that all the in-control points are with relatively high occurrence possibility. The advantages and effectiveness of the proposed control chart are demonstrated by some numerical examples in the end.     

O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N³) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N²)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener–Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N³) operations could be eliminated, and a typical speedup of 5–9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.     

A Modification on the Friedman Test Statistic

A modification on the well-known, nonparametric Friedman test statistic is suggested in this article. Probability distributions of the suggested test statistic under the null hypothesis are tabulated for some small sample cases. In addition to an example, simulation results for various sample sizes are presented. The simulation indicates that the modified test statistic performs better than the Friedman test in detecting treatment effects of small differences especially when the sample size is small.     

Comparisons of the percentage points of distributions with the same first four moments,chosen from eight different systems of frequency curves

Probability integrals and percentage points of univariate distributions from up to eight different families, having common first four moments are compared. Among interesting observations is the remarkable consistency in the standardized upper and lower 5% points over considerable regions of the √β₁, β₂ plane; also the closeness of agreement between the log-normal and non-central t distributions and the Pearson Type VI and Type IV curves respectively.     

Normalization rules for item response models: Simulation results

ABSTRACT

This research examines the statistical methodology that is used to estimate the parameters in item response models. An integral part of an item response model is the normalization rule that is used to identify the distributional parameters. The main result shown here is that only Verhelst–Glas normalizations that arbitrarily set one difficulty and one dispersion parameter to unity are consistent with the basic assumptions underlying the two-parameter logistic model. Failure to employ this type of normalization will lead to scores that depend on the item composition of the test and differential item difficulty (DIF) will compromise the validity of the estimated ability scores when different groups are being compared. It is also shown that some of the tests for DIF fail when the data are generated by an IRT model with a random effect. Most of the results are based on simulations of a four item model. Because the data generation mechanism is known, it is possible to determine the effect on ability scores and parameter estimates when different normalizations or different distribution parameter values are used.     

Dependence Structure of a Class of Symmetric Distributions

Abstract

In this article, dependence structure of a class of symmetric distributions is considered. Let X and Y be two n-dimensional random vectors having such distributions. We investigate conditions on the generators of densities of X and Y such that X is MTP₂, and X and Y can be compared in the multivariate likelihood ratio order. Nonnegativity of the covariance between functions of two adjacent order statistics of X is also given.     

Statistical Comparison of the Goodness of Fit Delivered by Five Families of Distributions Used in Distribution Fitting

A basic assumption in distribution fitting is that a single family of distributions may deliver useful representation to the universe of available distributions. To date, little study has been conducted to compare the relative effectiveness of these families. In this article, five families are compared by fitting them to a sample of 20 distributions, using 2 fitting objectives: minimization of the L ₂ norm and four-moment matching. Values of L ₂ norm associated with the fitted families are used as input data to test for significant differences. The Pearson family and the RMM (Response Modeling Methodology) family significantly outperforms all other families.     

An Asymptotic Characterization of Finite Degree U-statistics With Sample Size-Dependent Kernels: Applications to Nonparametric Estimators and Test Statistics

We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic’s kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric R² test for overall significance of a nonparametric regression model.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.