A Modification on the Friedman Test Statistic

A modification on the well-known, nonparametric Friedman test statistic is suggested in this article. Probability distributions of the suggested test statistic under the null hypothesis are tabulated for some small sample cases. In addition to an example, simulation results for various sample sizes are presented. The simulation indicates that the modified test statistic performs better than the Friedman test in detecting treatment effects of small differences especially when the sample size is small.     

Comparing Two Tests for Two Rates

This article rigorously proves superiority of the proportion χ² test to the logistic regression Wald test in terms of power when comparing two rates, despite their asymptotic equivalence under the null hypothesis that the two rates are equal.     

Small sample performance of large sample tests for tobit versus p -tobit

Large sample tests for the standard To bit model versus the ^p -Tobit model by Deaton and Irish (1984) are studied. The normalized one-tailed score test by Deaton and Irish (1984) is shown to be a version of Neyman's C(α) test that is valid for the non-standard problem of the null hypothesis lying on the boundary of the parameter space. Then, this paper reports the results of Monte Carlo experiments designed to study the small sample performance of large sample tests for the standard Tobit specification versus the ^p -Tobit specification.     

On Hotelling’s T2 test in a special paired sample case

In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.     

Generalized Analysis-of-variance-type Test for the Single-index Quantile Model

In this paper, we are concerned with a test for the index parameter and index function in the single-index model. Based on the estimates obtained by the quantile regression, we extend the generalized analysis-of-variance-type test to the single-index model. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows an asymptotically χ²-distribution. The simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods.     

Empirical-likelihood-based Test for Partially Linear Single-index Models with Error-prone Linear Covariates

In this article, we consider whether the empirical likelihood ratio (ELR) test is applicable to testing for serial correlation in the partially linear single-index models (PLSIM) with error-prone linear covariates. It is shown that under the null hypothesis the proposed ELR statistic follows asymptotically a χ²-distribution with the scale constant and the degrees of freedom. A comparison between the ELR and the normal approximation method is also considered. Both simulated and real data examples are used to illustrate our proposed methodology.     

Hayter and Tsui's test with double sampling for the vector mean of multivariate normal population

Abstract

In this paper, we introduce a version of Hayter and Tsui's statistical test with double sampling for the vector mean of a population under multivariate normal assumption. A study showed that this new test was more or as efficient than the well-known Hotelling's T² with double sampling. Some nice features of Hayter and Tsui's test are its simplicity of implementation and its capability of identifying the errant variables when the null hypothesis is rejected. Taking that into consideration, a new control chart called HTDS is also introduced as a tool to monitor multivariate process vector mean when using double sampling.     

Misspecified T2 tests. i. location and scale

Properties of Hotelling's (1931) T ² are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T ² and further properties of the T ² test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

On a test of independence for contingency tables

Using the concept of distributional distance, a test statistic is proposed FOR the hypothesis of independence in multidimensional contingency tables. A Monte Carlo Study is done to empirically compare the power of the proposed test to the Pearson x² and the likelihood ratio test- Further, the nonnull distribution under various spike alternatives is tabulated     

Testing homogeneity in discrete mixtures

This paper introduces W-tests for assessing homogeneity in mixtures of discrete probability distributions. A W-test statistic depends on the data solely through parameter estimators and, if a penalized maximum likelihood estimation framework is used, has a tractable asymptotic distribution under the null hypothesis of homogeneity. The large-sample critical values are quantiles of a chi-square distribution multiplied by an estimable constant for which we provide an explicit formula. In particular, the estimation of large-sample critical values does not involve simulation experiments or random field theory. We demonstrate that W-tests are generally competitive with a benchmark test in terms of power to detect heterogeneity. Moreover, in many situations, the large-sample critical values can be used even with small to moderate sample sizes. The main implementation issue (selection of an underlying measure) is thoroughly addressed, and we explain why W-tests are well-suited to problems involving large and online data sets. Application of a W-test is illustrated with an epidemiological data set.     

An Asymptotic Characterization of Finite Degree U-statistics With Sample Size-Dependent Kernels: Applications to Nonparametric Estimators and Test Statistics

We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic’s kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric R² test for overall significance of a nonparametric regression model.     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

A test for variance-covarianch parameters in normal linear models

This paper derives a test statistic for the variance-covariance parameters which is a quadratic function of their MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimates. The test is a Wald-type test, and its development closely parallels the theory used to derive a similar test for the coefficients in linear models. In fact, the derivation proceeds by first setting up the estimation problem in a derived linear model in which the dispersion parameters are the coefficients. The test statistic is shown to be the sum of the squares of independent standardized x² variables.     

Tests for Symmetry Based on the One-Sample Wilcoxon Signed Rank Statistic

ABSTRACT

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X ² statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.     

Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Hypothesis Testing of Hazard Ratio Parameters in Marginal Models for Multivariate Failure Time Data

Marginal hazard models for multivariate failure time data have been studied extensively in recent literature. However, standard hypothesis test statistics based on the likelihood method are not exactly appropriate for this kind of model. In this paper, extensions of the three commonly used likelihood hypothesis test statistics are discussed. Generalized Wald, generalized score and generalized likelihood ratio tests for hazard ratio parameters in a marginal hazard model for multivariate failure time data are proposed and their asymptotic distributions examined. The finite sample properties of these statistics are studied through simulations. The proposed method is applied to data from Busselton Population Health Surveys.     

Lack-of-fit testing of a regression model with response missing at random

This paper proposes a class of lack-of-fit tests for fitting a linear regression model when some response variables are missing at random. These tests are based on a class of minimum integrated square distances between a kernel type estimator of a regression function and the parametric regression function being fitted. These tests are shown to be consistent against a large class of fixed alternatives. The corresponding test statistics are shown to have asymptotic normal distributions under null hypothesis and a class of nonparametric local alternatives. Some simulation results are also presented.