Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

Sequential Monitoring Variance Changes in Location Model

In this paper, a moving monitoring procedure is proposed to detect potential variance change of the location model with dependent errors. The procedure is motivated by the problem that the existing square CUSUM test is insensitive to a late variance change of the location model. The asymptotic distribution of the statistics under the null hypothesis and the consistency under the alternative hypothesis are derived. Simulations show that our monitoring procedure compared to the square CUSUM test offers better power and can more quickly detect change. Moreover, the effectiveness of our procedure is illustrated by applying it to two data sets.     

A class of exceedance-type statistics for the two-sample problem

We consider here a class of test statistics based on exceeding observations and develop exceedance-type tests for the two-sample hypothesis testing problem. The exact distribution of the statistics are derived under the null hypothesis as well as under the Lehmann alternative, and then a comparative power study is carried out.     

Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Permuiation tesis for k-sample binomial data with comparisons of exact and approximate P-levels

Exact ksample permutation tests for binary data for three commonly encountered hypotheses tests are presented,, The tests are derived both under the population and randomization models . The generating function for the number of cases in the null distribution is obtained, The asymptotic distributions of the test statistics are derived . Actual significance levels are computed for the asymptotic test versions , Random sampling of the null distribution is suggested as a superior alternative to the asymptotics and an efficient computer technique for implementing the random sampling is described., finally, some numerical examples are presented and sample size guidelines given for computer implementation of the exact tests.     

Testing asymptotic independence in bivariate extremes

Bivariate extreme value condition (see (1.1) below) includes the marginal extreme value conditions and the existence of the (extreme) dependence function. Two cases are of interest: asymptotic independence and asymptotic dependence. In this paper, we investigate testing the existence of the dependence function under the null hypothesis of asymptotic independence and present two suitable test statistics. Small simulations are studied and the application for a real data is shown. The other case with the null hypothesis of asymptotic dependence is already investigated.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study     

Asymptotic distributions of some test statistics for a linear functional relationship

We consider the testing problems of the structural parameters for the multivariate linear functional relationship model. We treat the likelihood ratio test statistics and the test statistics based on the asymptotic distributions of the maximum likelihood estimators. We derive their asymptotic distributions under each null hypothesis respectively. A simulation study is made to evaluate how we can trust our asymptotic results when the sample size is rather small.     

Weighted Mean Survival Test Statistics: a Class of Distance Tests for Censored Survival Data

A class of test statistics is introduced which is sensitive against the alternative of stochastic ordering in the two-sample censored data problem. The test statistics for evaluating a cumulative weighted difference in survival distributions are developed while taking into account the imbalances in base-line covariates between two groups. This procedure can be used to test the null hypothesis of no treatment effect, especially when base-line hazards cross and prognostic covariates need to be adjusted. The statistics are semiparametric, not rank based, and can be written as integrated weighted differences in estimated survival functions, where these survival estimates are adjusted for covariate imbalances. The asymptotic distribution theory of the tests is developed, yielding test procedures that are shown to be consistent under a fixed alternative. The choice of weight function is discussed and relies on stability and interpretability considerations. An example taken from a clinical trial for acquired immune deficiency syndrome is presented.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.     

Likelihood ratio tests in linear mixed models with one variance component

Summary.  We consider the problem of testing null hypotheses that include restrictions on the variance component in a linear mixed model with one variance component and we derive the finite sample and asymptotic distribution of the likelihood ratio test and the restricted likelihood ratio test. The spectral representations of the likelihood ratio test and the restricted likelihood ratio test statistics are used as the basis of efficient simulation algorithms of their null distributions. The large sample χ ² mixture approximations using the usual asymptotic theory for a null hypothesis on the boundary of the parameter space have been shown to be poor in simulation studies. Our asymptotic calculations explain these empirical results. The theory of Self and Liang applies only to linear mixed models for which the data vector can be partitioned into a large number of independent and identically distributed subvectors. One-way analysis of variance and penalized splines models illustrate the results.     

An ANOVA-type test for multiple change points

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations. We propose an ANOVA-type test statistic and obtain its asymptotic null distribution. We also give approximations of its limiting critical values. We report the results of Monte Carlo studies conducted to compare the power of the proposed test against a number of its competitors. As illustrations we analyzed three real data sets.     

Asymptotic spacings theory with applications to the two-sample problem

The asymptotic distribution theory of test statistics which are functions of spacings is studied here. Distribution theory under appropriate close alternatives is also derived and used to find the locally most powerful spacing tests. For the two-sample problem, which is to test if two independent samples are from the same population, test statistics which are based on “spacing-frequencies” (i.e., the numbers of observations of one sample which fall in between the spacings made by the other sample) are utilized. The general asymptotic distribution theory of such statistics is studied both under the null hypothesis and under a sequence of close alternatives.     

协整参数的自举推断

 针对完全修正最小二乘(full-modified ordinary least square,简称FMOLS）估计方法,给出一种协整参数的自举推断程序,证明零假设下自举统计量与检验统计量具有相同的渐近分布。关于检验功效的研究表明,虽然有约束自举的实际检验水平表现良好,但如果零假设不成立,自举统计量的分布是不确定的,因而其经验分布不能作为检验统计量精确分布的有效估计。实际应用中建议使用无约束自举,因为无论观测数据是否满足零假设,其自举统计量与零假设下检验统计量都具有相同的渐近分布。最后,利用蒙特卡洛模拟对自举推断和渐近推断的有限样本表现进行比较研究。     

A new test for exponentiality against HNBUE alternatives

Abstract

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.     

Tests for a Multiple-Sample Problem in High Dimensions

In this article, we consider the problem of testing the hypothesis on mean vectors in multiple-sample problem when the number of observations is smaller than the number of variables. First we propose an independence rule test (IRT) to deal with high-dimensional effects. The asymptotic distributions of IRT under the null hypothesis as well as under the alternative are established when both the dimension and the sample size go to infinity. Next, using the derived asymptotic power of IRT, we propose an adaptive independence rule test (AIRT) that is particularly designed for testing against sparse alternatives. Our AIRT is novel in that it can effectively pick out a few relevant features and reduce the effect of noise accumulation. Real data analysis and Monte Carlo simulations are used to illustrate our proposed methods.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

On similarity and independence properties of composite edf goodness-of-fit tests

Many test statistics for classical simple goodness-of-fit hypothesis testing problems are distancemeasures between the distribution function of the null hypothesis distributipn and the empirical distribution function sometimes called EDF tests. If a composite parametric null hypothesis is considered in place of the simple null hypothesis, then a test statistic can be obtained from each EDF test by replacing the known distribution function of the simple problem by the Rao-Blackwell estimating distribution function. In this note we use known results to show that these Rao-Blackwell-EDF test statistics have distributions that do not depend upon parameter values, and hence that these tests are independent of a complete sufficient statistic for the parameters.