Diagonalization matrix and its application in distribution theory

Some matrix representations of diverse diagonal arrays are studied in this work; the results allow new definitions of classes of elliptical distributions indexed by kernels mixing Hadamard and usual products. A number of applications are derived in the setting of prior densities from the Bayesian multivariate regression model and families of non-elliptical distributions, such as the matrix multivariate generalized Birnbaum–Saunders density. The philosophy of the research about matrix representations of quadratic and inverse quadratic forms can be extended as a methodology for exploring possible new applications in non-standard distributions, matrix transformations and inference.     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.     

On singular elliptical models

The probability density function (pdf) ofsingular elliptical distributions is represented as an integralseries of singular normal distributions. Explicit formulas for the pdf and the cdf of the generalized Chi-square distribution are derived under singular elliptical assumptions extending the result of Díaz-García [(2002). Singular elliptical distribution: density and applications. Commun. Stat.—Theory Methods 31:665–681]. Applications are given of the proposed result for singular mixedmodels.     

On elliptical multilevel models

Multilevel models have been widely applied to analyze data sets which present some hierarchical structure. In this paper we propose a generalization of the normal multilevel models, named elliptical multilevel models. This proposal suggests the use of distributions in the elliptical class, thus involving all symmetric continuous distributions, including the normal distribution as a particular case. Elliptical distributions may have lighter or heavier tails than the normal ones. In the case of normal error models with the presence of outlying observations, heavy-tailed error models may be applied to accommodate such observations. In particular, we discuss some aspects of the elliptical multilevel models, such as maximum likelihood estimation and residual analysis to assess features related to the fitting and the model assumptions. Finally, two motivating examples analyzed under normal multilevel models are reanalyzed under Student-t and power exponential multilevel models. Comparisons with the normal multilevel model are performed by using residual analysis.     

Estimation in nonlinear mixed-effects models using heavy-tailed distributions

Nonlinear mixed-effects models are very useful to analyze repeated measures data and are used in a variety of applications. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. In this work, we introduce an extension of a normal nonlinear mixed-effects model considering a subclass of elliptical contoured distributions for both random effects and residual errors. This elliptical subclass, the scale mixtures of normal (SMN) distributions, includes heavy-tailed multivariate distributions, such as Student-t, the contaminated normal and slash, among others, and represents an interesting alternative to outliers accommodation maintaining the elegance and simplicity of the maximum likelihood theory. We propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed-effects and variance components, using a stochastic approximation of the EM algorithm. We compare the performance of the normal and the SMN models with two real data sets.     

An Affine-Invariant Generalization of the Wilcoxon Signed-Rank Test for the Bivariate Location Problem

This paper proposes an affine‐invariant test extending the univariate Wilcoxon signed‐rank test to the bivariate location problem. It gives two versions of the null distribution of the test statistic. The first version leads to a conditionally distribution‐free test which can be used with any sample size. The second version can be used for larger sample sizes and has a limiting χ₂² distribution under the null hypothesis. The paper investigates the relationship with a test proposed by Jan & Randles (1994). It shows that the Pitman efficiency of this test relative to the new test is equal to 1 for elliptical distributions but that the two tests are not necessarily equivalent for non‐elliptical distributions. These facts are also demonstrated empirically in a simulation study. The new test has the advantage of not requiring the assumption of elliptical symmetry which is needed to perform the asymptotic version of the Jan and Randles test.     

A family of bivariate distributions with non-elliptical contours

In this paper, a family of copulas with two parameters is proposed and its dependence analysis is performed. The corresponding family of bivariate distributions with specified marginals is constructed. For normal marginals, the new distributions are non-elliptical and can be applied in data analysis. They provide various alternative hypotheses for testing normality. Finally, an example is given.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Estimation and Testing in Elliptical Functional Measurement Error Models

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.     

A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal–Wallis (K–W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K–W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K–W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.     

TESTING THE AGREEMENT OF TWO QUANTITATIVE ASSAYS IN INDIVIDUAL MEANS

ABSTRACT

Existing approaches for the statistical evaluation of the agreement of two quantitative assays in terms of individual means are either based on a linear model and some stringent assumptions or comparisons of averages of individual means. Furthermore, the related statistical tests for some of these approaches are not valid in the sense that the sizes of these tests are not exactly the same as the nominal size even asymptotically. In this paper we propose a new method, which produces exact statistical tests that are easy to compute. When independent replicates are available, the proposed method requires very little or no assumption on the individual error variances. Simulation results show that the proposed tests perform better than some existing tests. Some examples are presented for illustration.     

On optimal tests for general interval-hypotheses

Optimal statistical tests, using the normality assumptions for general interval hypotheses including equivalence testing and testing for nonzero difference (or for non-unit) are presented. These tests are based on the decision theory for Polya Type distributions and are compared with usual confidence tests and with ’two one-sided tests’- procedures. A formal relationship between some optimal tests and the Anderson and Hauck procedure as well as a procedure recommended by Patel and Gupta is given. A new procedure for a generalisation of Student's test as well as for equivalence testing for thet-statistics is shown.     

BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.     

UNIT ROOT TESTING IN THE PRESENCE OF HEAVY‐TAILED GARCH ERRORS

We derive the asymptotic distributions of the Dickey–Fuller (DF) and augmented DF (ADF) tests for unit root processes with Generalized Autoregressive Conditional Heteroscedastic (GARCH) errors under fairly mild conditions. We show that the asymptotic distributions of the DF tests and ADF t‐type test are the same as those obtained in the independent and identically distributed Gaussian cases, regardless of whether the fourth moment of the underlying GARCH process is finite or not. Our results go beyond earlier ones by showing that the fourth moment condition on the scaled conditional errors is totally unnecessary. Some Monte Carlo simulations are provided to illustrate the finite‐sample‐size properties of the tests.     

Some useful integrals and their applications in correlation analysis

The distribution of the product moment correlation coefficient based on the bivariate normal distribution is well known. Recently in many business and economic data, fat tailed distributions especially some elliptical distributions have been considered as parent populations. The normal and t-distributions are well known special cases of elliptical distribution. In this paper we derive some theorems involving double integrals and apply them to derive the probability distribution of the correlation coefficient for some elliptical populations. The general nature of the theorems indicates their potential use in probability distribution theory.     

Variable selection in elliptical linear mixed model

Variable selection in elliptical Linear Mixed Models (LMMs) with a shrinkage penalty function (SPF) is the main scope of this study. SPFs are applied for parameter estimation and variable selection simultaneously. The smoothly clipped absolute deviation penalty (SCAD) is one of the SPFs and it is adapted into the elliptical LMM in this study. The proposed idea is highly applicable to a variety of models which are set up with different distributions such as normal, student-t, Pearson VII, power exponential and so on. Simulation studies and real data example with one of the elliptical distributions show that if the variable selection is also a concern, it is worthwhile to carry on the variable selection and the parameter estimation simultaneously in the elliptical LMM.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Abstract

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.