Improved tests for homogeneity of variances

Equality of variances is one of the key assumptions of analysis of variances (ANOVA). There are several testing procedures available to validate this assumption, but it is rare to find a test procedure which controls the type I error rate while providing high statistical power. In this article, we introduce a bootstrap test based on the ratio of mean absolute deviances (RMD). We also propose a two-stage testing procedure where we first quantify the skewness of the distributions and then choose an appropriate test for homogeneity of variances. The performances of these test procedures are studied via a simulation study.     

A simultaneous variable selection methodology for linear mixed models

Selecting an appropriate structure for a linear mixed model serves as an appealing problem in a number of applications such as in the modelling of longitudinal or clustered data. In this paper, we propose a variable selection procedure for simultaneously selecting and estimating the fixed and random effects. More specifically, a profile log-likelihood function, along with an adaptive penalty, is utilized for sparse selection. The Newton-Raphson optimization algorithm is performed to complete the parameter estimation. By jointly selecting the fixed and random effects, the proposed approach increases selection accuracy compared with two-stage procedures, and the usage of the profile log-likelihood can improve computational efficiency in one-stage procedures. We prove that the proposed procedure enjoys the model selection consistency. A simulation study and a real data application are conducted for demonstrating the effectiveness of the proposed method.     

Subset selection in two-factor experiments using randomization restricted designs

This paper studies subset selection procedures for screening in two-factor treatment designs that employ either a split-plot or strip-plot randomization restricted experimental design laid out in blocks. The goal is to select a subset of treatment combinations associated with the largest mean. In the split-plot design, it is assumed that the block effects, the confounding effects (whole-plot error) and the measurement errors are normally distributed. None of the selection procedures developed depend on the block variances. Subset selection procedures are given for both the case of additive and non-additive factors and for a variety of circumstances concerning the confounding effect and measurement error variances. In particular, procedures are given for (1) known confounding effect and measurement error variances (2) unknown measurement error variance but known confounding effect (3) unknown confounding effect and measurement error variances. The constants required to implement the procedures are shown to be obtainable from available FORTRAN programs and tables. Generalization to the case of strip-plot randomization restriction is considered.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

One-sided range test for testing against an ordered alternative under heteroscedasticity

In a one-way fixed effects analysis of variance model, when normal variances are unknown and possibly unequal, a one-sided range test for testing the null hypothesis H ₀ : μ ₁ = … = μ_k against an ordered alternative H_a : μ ₁ ≤ … ≤ μ_k by a single-stage and a two-stage procedure, respectively, is proposed. The critical values under H ₀ and the power under a specific alternative are calculated. Relation between the one-stage and the two-stage test procedures is discussed. A numerical example to illustrate these procedures is given.     

A Multiple Three-Decision Procedures for Comparing Several Treatments with a Control Under Heteroscedasticity

In this article, the design-oriented two-stage multiple three-decision procedure is proposed to classify a set of normal populations with respect to a control under heteroscedasticity. The statistical tables of percentage points and the power-related design constants, to implement our new two-stage procedure, are given. Sometimes when the sample for the second stage is not available, the one-stage data analysis procedure is proposed. Classifying a treatment better than control when it is actually worse (and vice versa) is known as type III error. Both the two-stage and one-stage procedures control the type III error rate at a specified level. The relationship between the two-stage and one-stage procedures is discussed. Finally, the application of the proposed procedures is illustrated with an example.     

A note on homogeneity tests of covariance matrices

In this note we propose two procedures for testing homogeneity of co-variance matrices that are both extensions of Hartley's (1940) test for equality of variances. The first is a two-stage procedure where the first step is a simple test for equality of the largest eigenvalues, and corresponding eigenvectors, of the covariance matrices. The second is based on projection pursuit and seems harder to apply in practice.     

On the Skew-Normal-Cauchy Distribution

A fixed effects one-way layout model of analysis of variance is considered where the variances are taken to be possibly unequal. Conservative single-stage procedures based on Banerjee’s method for the solution of the Behrens-Fisher problem are proposed for the following multiple comparisons problems: 1) all pairwise comparisons with a control population mean, and 2) all pairwise comparisons and all linear contrasts among the means. Since these procedures are likely to be very conservative in practice, approximate procedures based on Welch’s method for the solution of the Behrens-Fisher problem are suggested as alternatives. Monte Carlo studies indicate that the latter are much less conservative and hence may be better in practice. Both these sets of procedures need only the tables of the Student’s t-distribution for their application and are very simple to use. Exact two-stage procedures are proposed for the following multiple comparisons problems: 1) all pairwise comparisons and all linear contrasts among the means, and 2) all linear combinations of the means.     

Selection of a better component in bivariate exponential models

Summary Selection procedures of the better component in bivariate exponential (BVE) models are proposed. In this paper, we consider onlyBVE models proposed by Freund (1961) Marshall-Olkin (1967) and Block-Basu (1974). The probabilities of correct selection for the proposed procedures are compared by using the normal approximations. A numerical study on the determination of asymptotic relative efficiency (ARE) of the proposed procedures are presented.     

Modeling statistical dependence of Markov chains via copula models

Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set.     

A Class of Two-Stage Selection Procedures Using L-Statistics

Summary: A class of selection procedures for selecting the least dispersive distribution from k available distributions has been proposed. This problem finds applications in reliability and engineering. In engineering, for example, the goal of the experimenter is to select a firm whose components have least dispersive distribution from the available set of competing firms manufacturing the components of the desired specifications meant for the same purpose. The proposed procedures can be used even when the underlying distributions belong to different families. Applications of the proposed selection procedures are discussed by taking exponential, gamma and Lehmann type distributions. Performance of the proposed selection procedures is assessed through simulation study. Implementation of the proposed selection procedure is illustrated through an example. * The authors are very grateful to the editor and referees for their valuable comments.     

An improved two-stage selection procedure

The problem of selecting the best of k normal populations with unknown and possibly unequal variances is considered The two-stage procedure proposed by Rinott (1978) is improved so that less samples need to be drawn in the second-stage of the sampling scheme     

Model selection of generalized partially linear models with missing covariates

In this paper, a generalized partially linear model (GPLM) with missing covariates is studied and a Monte Carlo EM (MCEM) algorithm with penalized-spline (P-spline) technique is developed to estimate the regression coefficients and nonparametric function, respectively. As classical model selection procedures such as Akaike's information criterion become invalid for our considered models with incomplete data, some new model selection criterions for GPLMs with missing covariates are proposed under two different missingness mechanism, say, missing at random (MAR) and missing not at random (MNAR). The most attractive point of our method is that it is rather general and can be extended to various situations with missing observations based on EM algorithm, especially when no missing data involved, our new model selection criterions are reduced to classical AIC. Therefore, we can not only compare models with missing observations under MAR/MNAR settings, but also can compare missing data models with complete-data models simultaneously. Theoretical properties of the proposed estimator, including consistency of the model selection criterions are investigated. A simulation study and a real example are used to illustrate the proposed methodology.     

New procedures for selecting the k-best exponential populations

Suppose exponential populations π_i with parameters (μ_i,σ_i) (i = 1, 2, …, K) are given. The σ_i can be unknown and unequal. This article discusses how to select the k (≥1) best populations. Under the subset selection formulation, a one-stage procedure is proposed. Under the indifference zone formulation, a two-stage procedure is proposed. An appealing feature of these procedures is that no statistical tables are needed for their implementation.     

Some comments on two-stage selection procedures

In this paper we study the procedures of Dudewicz and Dalal ( 1975 ), and the modifications suggested by Rinott ( 1978 ), for selecting the largest mean from k normal populations with unknown variances. We look at the case k = 2 in detail, because there is an optimal allocation scheme here. We do not really allocate the total number of samples into two groups, but we estimate this optimal sample size, as well, so as to guarantee the probability of correct selection (written as P(CS)) at least P^?, 1/2 < P^? < 1 . We prove that the procedure of Rinott is “asymptotically in-efficient” (to be defined below) in the sense of Chow and Robbins ( 1965 ) for any k  2. Next, we propose two-stage procedures having all the properties of Rinott's procedure, together with the property of “asymptotic efficiency” - which is highly desirable.     

Focused Information Criterion and Model Averaging in Quantile Regression

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice.     

Adaptive LASSO for linear mixed model selection via profile log-likelihood

Mixed model selection is quite important in statistical literature. To assist the mixed model selection, we employ the adaptive LASSO penalized term to propose a two-stage selection procedure for the purpose of choosing both the random and fixed effects. In the first stage, we utilize the penalized restricted profile log-likelihood to choose the random effects; in the second stage, after the random effects are determined, we apply the penalized profile log-likelihood to select the fixed effects. In each stage, the Newton–Raphson algorithm is performed to complete the parameter estimation. We prove that the proposed procedure is consistent and possesses the oracle properties. The simulations and a real data application are conducted for demonstrating the effectiveness of the proposed selection procedure.     

Subset Selection Toward Optimizing the Best Performance at a Second Stage

In the search for the best of n candidates, two-stage procedures of the following type are in common use. In a first stage, weak candidates are removed, and the subset of promising candidates is then further examined. At a second stage, the best of the candidates in the subset is selected. In this article, optimization is not aimed at the parameter with largest value but rather at the best performance of the selected candidates at Stage 2. Under a normal model, a new procedure based on posterior percentiles is derived using a Bayes approach, where nonsymmetric normal (proper and improper) priors are applied. Comparisons are made with two other procedures frequently used in selection decisions. The three procedures and their performances are illustrated with data from a recent recruitment process at a Midwestern university.