CRITICAL VALUE APPROXIMATIONS FOR TESTS OF LINEAR REGRESSION DISTURBANCES

Two important classes of tests for non-spherical disturbances in the linear regression model involve test statistics whose null distributions and hence critical values depend on the regressors. This paper investigates the accuracy of the normal, two moment beta and four moment beta approximations to the critical values of such tests. An empirical experiment aimed at evaluating the accuracy of the approximations for a variety of tests against autocorrelation and heteroscedasticity is conducted. Overall the approximations are found to provide reasonably accurate critical values with skewness being a factor determining the degree of accuracy.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

THE OPTIMAL CRITICAL VALUE OF A PRE-TEST FOR AN INEQUALITY RESTRICTION IN A MIS-SPECIFIED REGRESSION MODEL

Optimal critical values are derived for a pre-test of an inequality restriction in a model where relevant regressors are unwittingly omitted. The criterion adopted is either that of the minimum average relative risk, or that of the mini-max regret. The latter approach yields an optimal critical value which is sensitive to the degree of model mis-specification, while the former criterion always leads to the choice of the unrestricted estimator.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

THE POWER AND SIZE OF NONPARAMETRIC TESTS FOR COMMON DISTRIBUTIONAL CHARACTERISTICS

This paper considers the power and size properties of some well known nonparametric linear rank tests for location and scale as well as the Kolmogorov-Smirnov omnibus test and proposed alternatives to it. Independence between some classes of linear rank tests is established facilitating their joint application. Monte Carlo study confirms the asymptotic power properties of the linear rank tests but raises concerns about their application in more general and practically relevant circumstances. It also indicates that the new omnibus tests constitute viable alternatives with superior properties to the Kolmogorov-Smirnov test in certain circumstances.     

SIMULATIONS AND DERIVED APPROXIMATIONS FOR THE MEANS AND STANDARD DEVIATIONS OF THE CHARACTERISTIC ROOTS OF A WISHART MATRIX

Monte Carlo simulations were done to estimate the means and standard deviations of the characteristic roots of a Wishart matrix which can be used in computing tests of hypotheses concerning multiplicative terms in balanced linear-bilinear (multiplicative) models for an m × n table of data. In this report we extend the previous results (Mandel, 1971; Cornelius, 1980) to r ≤ 199, c ≤ 149 or r ≤ 149, c ≤ 199, where r and c are row and column degrees of freedom, respectively, of the two-way array of residuals (with total degrees of freedom rc) after fitting the linear effects. For 187 combinations of r and c at intervals over this domain, we used 5000 simulations to estimate expectations and standard deviations of the Wishart roots. Using weighted linear regression variable selection techniques, symmetric functions of r and c were obtained for approximating the simulated means and standard deviations. Use of these approximating functions will avoid the need for reference to tables for input to computer programs which require these values for tests of significance of sequentially fitted terms in the analyses of balanced linear-bilinear models.     

OPTIMALITY OF NONPARAMETRIC TESTS FOR ASSOCIATION BETWEEN SURVIVAL TIME AND CONTINUOUS COVARIATES

Abstract

Recently, Chen (Chen, Z. (2000 Chen, Z. 2000. A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters, 49: 155–161. [Crossref], [Web of Science ®] , [Google Scholar]). A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters 49:155–161.) proposed a two-parameter model that can be used to model bathtub-shaped failure rate. Although this model has several interesting properties, it does not contain a scale parameter and hence not flexible in modeling real data. A generalized model including the scale parameter has shown to be interesting and it has the traditional Weibull distribution as an asymptotic case. In this article, a detailed analysis of this model is presented. Shapes of the density and failure rate function are studied. The asymptotic confidence intervals for the parameters are also derived from the Fisher information matrix. The likelihood ratio test is applied to test the goodness of fit of Weibull extension model. Some examples are shown to illustrate the application of the model and analysis.     

A MONTE CARLO COMPARISON OF ELLIPTICAL-THEORY AND OTHER TESTS FOR EQUALITY OF COVARIANCE MATRICES

A Monte Carlo study of the size and power of tests of equality of two covariance matrices is carried out. Tests based upon normality assumptions, elliptical distribution assumptions as well as distribution-free tests are compared. Samples are generated from normal, elliptical and non-elliptical populations. The elliptical-theory tests, in particular, have poor size properties for both elliptical distributions with moderate sample sizes and for non-elliptical distributions.     

SIZE CHARACTERISTICS OF TESTS FOR SAMPLE SELECTION BIAS: A MONTE CARLO COMPARISON AND EMPIRICAL EXAMPLE

The t-test of an individual coefficient is used widely in models of qualitative choice. However, it is well known that the t-test can yield misleading results when the sample size is small. This paper provides some experimental evidence on the finite sample properties of the t-test in models with sample selection biases, through a comparison of the t-test with the likelihood ratio and Lagrange multiplier tests, which are asymptotically equivalent to the squared t-test. The finite sample problems with the t-test are shown to be alarming, and much more serious than in models such as binary choice models. An empirical example is also presented to highlight the differences in the calculated test statistics.     

PERMUTATION TESTS AND OTHER TEST STATISTICS FOR ILL-BEHAVED DATA: EXPERIENCE OF THE NINDS t-PA STROKE TRIAL

Permutation tests are often used to analyze data since they may not require one to make assumptions regarding the form of the distribution to have a random and independent sample selection. We initially considered a permutation test to assess the treatment effect on computed tomography lesion volume in the National Institute of Neurological Disorders and Stroke (NINDS) t-PA Stroke Trial, which has highly skewed data. However, we encountered difficulties in summarizing the permutation test results on the lesion volume. In this paper, we discuss some aspects of permutation tests and illustrate our findings. This experience with the NINDS t-PA Stroke Trial data emphasizes that permutation tests are useful for clinical trials and can be used to validate assumptions of an observed test statistic. The permutation test places fewer restrictions on the underlying distribution but is not always distribution-free or an exact test, especially for ill-behaved data. Quasi-likelihood estimation using the generalized estimating equation (GEE) approach on transformed data seems to be a good choice for analyzing CT lesion data, based on both its corresponding permutation test and its clinical interpretation.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.