Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Alternative boundaries for CUSUM tests

Alternative boundaries for the common Recursive (or Standard) CUSUM test and the OLS-based CUSUM test for structural change are suggested and their properties are examined by simulation of expectedp values. The poor power of the tests for early and late structural changes can be improved for the OLS-based version, whereas this weakness of the Recursive CUSUM test cannot be overcome by the new boundaries. Research supported by the Austrian Science Foundation (FWF) under grant SFB#010 (‘Adaptive Information Systems and Modeling in Economics and Management Science’).     

Comparisons of Test Statistics Arising from Marginal Analyses of Multivariate Survival Data

We investigate the properties of several statistical tests for comparing treatment groups with respect to multivariate survival data, based on the marginal analysis approach introduced by Wei, Lin and Weissfeld [Regression Analysis of multivariate incomplete failure time data by modelling marginal distributians, JASA vol. 84 pp. 1065–1073]. We consider two types of directional tests, based on a constrained maximization and on linear combinations of the unconstrained maximizer of the working likelihood function, and the omnibus test arising from the same working likelihood. The directional tests are members of a larger class of tests, from which an asymptotically optimal test can be found. We compare the asymptotic powers of the tests under general contiguous alternatives for a variety of settings, and also consider the choice of the number of survival times to include in the multivariate outcome. We illustrate the results with simulations and with the results from a clinical trial examining recurring opportunistic infections in persons with HIV.     

ASSESSING CONGRUENCEAMONG DISTANCE MATRICES: SINGLE-MALT SCOTCH WHISKIES REVISITED

A test of congruence among distance matrices is described. It tests the hypothesis that several matrices, containing different types of variables about the same objects, are congruent with one another, so they can be used jointly in statistical analysis. Raw data tables are turned into similarity or distance matrices prior to testing; they can then be compared to data that naturally come in the form of distance matrices. The proposed test can be seen as a generalization of the Mantel test of matrix correspondence to any number of distance matrices. This paper shows that the new test has the correct rate of Type I error and good power. Power increases as the number of objects and the number of congruent data matrices increase; power is higher when the total number of matrices in the study is smaller. To illustrate the method, the proposed test is used to test the hypothesis that matrices representing different types of organoleptic variables (colour, nose, body, palate and finish) in single‐malt Scotch whiskies are congruent.     

Signed likelihood ratio tests in the Birnbaum-Saunders regression model

The Birnbaum-Saunders regression model is becoming increasingly popular in lifetime analyses and reliability studies. In this model, the signed likelihood ratio statistic provides the basis for testing inference and construction of confidence limits for a single parameter of interest. We focus on the small sample case, where the standard normal distribution gives a poor approximation to the true distribution of the statistic. We derive three adjusted signed likelihood ratio statistics that lead to very accurate inference even for very small samples. Two empirical applications are presented.     

A ratio-to-control Williams-type test for trend

Under certain conditions, many multiple contrast tests based on the difference of treatment means can also be conveniently expressed in terms of ratios. In this paper, a Williams test for trend is defined as ratios-to-control for ease of interpretation and to obtain directly comparable confidence intervals. Simultaneous confidence intervals for percentages are particularly helpful for interpretations in the case of multiple endpoints. Methods for constructing simultaneous confidence intervals are discussed under both homogeneous and heterogeneous error variances. This approach is available in the R extension package mratios. The proposed method is used to test for trend in an immunotoxicity study with several endpoints as an example.     

线性回归模型Bootstrap LM-Lag检验有效性研究

基于OLS估计残差,将Bootstrap方法用于空间滞后相关LM-Lag检验。在不同的误差结构和空间权重矩阵条件下,比较Bootstrap LM-Lag检验和渐近检验的水平扭曲和功效。通过Monte Carlo实验表明,当误差项不服从经典正态分布假设时,LM-Lag渐近检验存在严重的水平扭曲,Bootstrap检验能够有效地校正水平扭曲,并且Bootstrap LM-Lag检验的功效与渐近检验近似;无论误差项是否服从正态分布,从水平扭曲和功效角度看,线性回归模型Bootstrap LM-Lag检验有效。