Asymptotic optimality of multivariate linear hypothesis tests

The optimal exponential rate at which the Type II error probability of a multivariate linear hypothesis test can tend to zero while the Type I error probability is held fixed is given. The likelihood ratio test, the test of Hotelling and Lawley, the test of Bartlett, Nanda, and Pillai, and the test of Roy are shown to be asymptotically optimal in the sense that for each of these tests the exponential rate of convergence of the type II error probability attains the optimal value. Some other tests for the multivariate linear hypothesis are shown not to be asymptotically optimal.     

Some test statistics for the structural coefficients of the multivariate linear functional relationship model

Summary For the testing problem concerning the coefficients of the multivariate linear functional relationship model, the distribution of a statistic previously proposed by A. P. Basu depends on the unknown covariance matrixV of errors, so limiting its applicability. This article proposes new test statistics with sampling distributions which are independent of the unknown parameters for the cases whereV is either unknown or known only up to a proportionality factor. The exact distributions of the test statistics are also discussed.     

Latent scale linear models for multivariate ordinal responses and analysis by the method of weighted least squares

Summary A multivariate latent scale linear model is defined for multivariate ordered categorical responses and inference procedures based on the weighted least squares method are developed. Several applications of the model are suggested and illustrated through an analysis of real data. Asymptotic properties of the weighted least squares method are examined and some consequences of misspecification of the model are also discussed.     

Superheat: An R Package for Creating Beautiful and Extendable Heatmaps for Visualizing Complex Data

The technological advancements of the modern era have enabled the collection of huge amounts of data in science and beyond. Extracting useful information from such massive datasets is an ongoing challenge as traditional data visualization tools typically do not scale well in high-dimensional settings. An existing visualization technique that is particularly well suited to visualizing large datasets is the heatmap. Although heatmaps are extremely popular in fields such as bioinformatics, they remain a severely underutilized visualization tool in modern data analysis. This article introduces superheat, a new R package that provides an extremely flexible and customizable platform for visualizing complex datasets. Superheat produces attractive and extendable heatmaps to which the user can add a response variable as a scatterplot, model results as boxplots, correlation information as barplots, and more. The goal of this article is two-fold: (1) to demonstrate the potential of the heatmap as a core visualization method for a range of data types, and (2) to highlight the customizability and ease of implementation of the superheat R package for creating beautiful and extendable heatmaps. The capabilities and fundamental applicability of the superheat package will be explored via three reproducible case studies, each based on publicly available data sources.     

Similar tests for covariance structures in multivariate linear models

Nyblom (J. Multivariate Anal. 76 (2001) 294) has derived locally best invariant test for the covariance structure in a multivariate linear model. The class of invariant tests obtained by Nyblom [9] does not coincide with the class of similar tests for this testing set-up. This paper extends some of the results of Nyblom [9] by deriving the locally best similar tests for the covariance structure. Moreover, it develops a saddlepoint approximation to optimal weighted average power similar tests (i.e. tests which maximize a weighted average power).     

Local asymptotic normality of multivariate ARMA processes with a linear trend

The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linear models with ARMA error term. In contrast with earlier univariate results, the central sequence here is correlogram-based, i.e. expressed in terms of a generalized concept of residual cross-covariance function.     

Bayesian hierarchical linear mixed models for additive smoothing splines

Bayesian hierarchical models have been used for smoothing splines, thin-plate splines, and L-splines. In analyzing high dimensional data sets, additive models and backfitting methods are often used. A full Bayesian analysis for such models may include a large number of random effects, many of which are not intuitive, so researchers typically use noninformative improper or nearly improper priors. We investigate propriety of the posterior for these cases. Our findings extend known results for normal linear mixed models to certain cases with Bayesian additive smoothing spline models. Supported by National Science Foundation grant SES-0351523 and by National Institutes of Health grants R01-CA100760 and R01-MH071418.     

Reduced-rank regression for the multivariate linear model

The problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed. We show that this problem is related to the problem of deciding how many principal components or pairs of canonical variates to use in any practical situation. Under the assumption of joint normality of the two sets of variates, we give the asymptotic (large-sample) distributions of the various estimated reduced-rank regression coefficient matrices that are of interest. Approximate confidence bounds on the elements of these matrices are then suggested using either the appropriate asymptotic expressions or the jackknife technique.     

Testing goodness of fit for the distribution of errors in multivariate linear models

In this paper, to test goodness of fit to any fixed distribution of errors in multivariate linear models, we consider a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis. We study the limiting behaviour of this test statistic under the null hypothesis and under alternatives. In the asymptotics, the rank of the design matrix is allowed to grow with the sample size.     

The superiorities of bayes linear unbiased estimator in multivariate linear models

In this article,the Bayes linear unbiased estimator (BALUE) of parameters is derived for the multivariate linear models.The superiorities of the BALUE over the least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Bayesian Pitman closeness (PC) criterion.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

An MOLP based procedure for finding efficient units in DEA models

In this paper a multiple objective linear programming (MOLP) problem whose feasible region is the production possibility set with variable returns to scale is proposed. By solving this MOLP problem by multicriterion simplex method, the extreme efficient Pareto points can be obtained. Then the extreme efficient units in data envelopment analysis (DEA) with variable returns to scale, considering the specified theorems and conditions, can be obtained. Therefore, by solving the proposed MOLP problem, the non-dominant units in DEA can be found. Finally, a numerical example is provided.     

A complete class for linear estimation in a general linear model

Summary It is shown that in linear estimation both unbiased and biased, all unique (up to equivalence with respect to risk) locally best estimators and their limits constitute a complete class.     

Generalized linear models for the analysis of taguchi-type experiments

Recent interest in Taguchi's methods have led to developments in joint analysis of the mean and dispersion from designed experiments. A commonly used method is the analysis of variance of the transformed data. However, a single transformation cannot necessarily produce the Normality, constancy of variance and linearity of systematic effects for the mean and dispersion models. We describe the use of generalized linear models for the analysis of such experiments and illustrate the methods with a data set.     

Empirical likelihood for partial linear models

In this paper the empirical likelihood method due to Owen (1988,Biometrika,75, 237–249) is applied to partial linear random models. A nonparametric version of Wilks' theorem is derived. The theorem is then used to construct confidence regions of the parameter vector in the partial linear models, which has correct asymptotic coverage. A simulation study is conducted to compare the empirical likelihood and normal approximation based method. Research supported by NNSF of China and a grant to the first author for his excellent Ph.D. dissertation work in China. Research supported by Hong Kong RGC CERG No. HKUST6162/97P.     

线性纵向数据模型中多个个体的联合影响诊断

本文利用个体删除方法对具有纯序列相关的线性纵向数据模型,给出了多个个体对参数联合影响的分析式,并将其化简成相对容易计算的形式,同时讨论了enhancing、reducing及swamping效应.进一步,分析了个人所得税申报数据,发现了单个个体删除方法无法识别的影响个体,验证了多个个体删除方法在寻找影响个体时的有效性,扩大了删除方法的应用领域.     

Robustness of F-tests in singular linear models

F-test is the most popular test in the general linear model. However, there is few discussions on the robustness of F-test under the singular linear model. In this paper, the necessary and sufficient conditions of robust F-test statistic are given under the general linear models or their partition models, which allows that the design matrix has deficient rank and the covariance matrix of error is a nonnegative definite matrix with parameters. The main results obtained in this paper include the existing findings of the general linear model under the definite covariance matrix. The usage of the theorems is illustrated by an example.     

The asymptotic validity of invariant procedures for the repeated measures model and multivariate linear model

Fairly general sufficient conditions are given to guarantee that invariant tests about means in the multivariate linear model and the repeated measures model have the correct asymptotic size when the normal assumption under which the tests are derived is relaxed. These conditions are the same as Huber's condition which guarantees asymptotic validity of the size of the F-test for the univariate linear model.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.