WILCOXON-TYPE RANK-SUM PRECEDENCE TESTS

This paper introduces Wilcoxon‐type rank‐sum precedence tests for testing the hypothesis that two life‐time distribution functions are equal. They extend the precedence life‐test first proposed by Nelson in 1963. The paper proposes three Wilcoxon‐type rank‐sum precedence test statistics—the minimal, maximal and expected rank‐sum statistics—and derives their null distributions. Critical values are presented for some combinations of sample sizes, and the exact power function is derived under the Lehmann alternative. The paper examines the power properties of the Wilcoxon‐type rank‐sum precedence tests under a location‐shift alternative through Monte Carlo simulations, and it compares the power of the precedence test, the maximal precedence test and Wilcoxon rank‐sum test (based on complete samples). Two examples are presented for illustration.     

Consistent testing for non‐correlation of two cointegrated ARMA time series

A consistent approach to the problem of testing non‐correlation between two univariate infinite‐order autoregressive models was proposed by Hong (1996). His test is based on a weighted sum of squares of residual cross‐correlations, with weights depending on a kernel function. In this paper, the author follows Hong's approach to test non‐correlation of two cointegrated (or partially non‐stationary) ARMA time series. The test of Pham, Roy & Cédras (2003) may be seen as a special case of his approach, as it corresponds to the choice of a truncated uniform kernel. The proposed procedure remains valid for testing non‐correlation between two stationary invertible multivariate ARMA time series. The author derives the asymptotic distribution of his test statistics under the null hypothesis and proves that his procedures are consistent. He also studies the level and power of his proposed tests in finite samples through simulation. Finally, he presents an illustration based on real data.     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.     

Precedence tests for equality of two distributions based on early failures of ranked set samples

In this article, two different types of precedence tests, each with two different test statistics, based on ranked set samples for testing the equality of two distributions are discussed. The exact null distributions of proposed test statistics are derived, critical values are tabulated for both set size and number of cycles up to 8, and the exact power functions of these two types of precedence tests under the Lehmann alternative are derived. Then, the power values of these two test procedures and their competitors based on simple random samples and based on ranked set samples are compared under the Lehmann alternative exactly and also under a location-shift alternative by means of Monte Carlo simulations. Finally, the impact of imperfect ranking is discussed and some concluding remarks are presented.     

Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data.     

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.     

Tests for time series of counts based on the probability-generating function

We propose testing procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law. The new test statistics incorporate the empirical probability-generating function computed from the observations. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is included as well as real-data examples.     

Multivariate hypothesis testing using generalized and {2}-inverses – with applications

The use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions. In this article, the use of {2} -inverses for that problem is investigated. Alternatively, Imhof-based test statistics can also be defined, which converge in distribution to weighted sum of chi-square variables. The asymptotic distributions of these test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.     

A weighted bootstrap approximation for comparing the error distributions in nonparametric regression

Several procedures have been proposed for testing the equality of error distributions in two or more nonparametric regression models. Here we deal with methods based on comparing estimators of the cumulative distribution function (CDF) of the errors in each population to an estimator of the common CDF under the null hypothesis. The null distribution of the associated test statistics has been approximated by means of a smooth bootstrap (SB) estimator. This paper proposes to approximate their null distribution through a weighted bootstrap. It is shown that it produces a consistent estimator. The finite sample performance of this approximation is assessed by means of a simulation study, where it is also compared to the SB. This study reveals that, from a computational point of view, the proposed approximation is more efficient than the one provided by the SB.     

TESTS FOR SEASONAL DIFFERENCING WITH AN UNKNOWN BREAK-POINT

Some Lagrange multiplier tests for seasonal differencing are proposed; their main objective is to avoid over-differencing due to structural change. The null hypothesis is either the presence of both regular and seasonal unit roots or the presence of a seasonal unit root. Alternative hypotheses allow for stationarity around a possible structural change where the break-point is unknown. The location of the structural change is estimated using the proposed procedures, the asymptotic distribution of the test statistics under the null hypothesis is derived and some useful percentiles are tabulated. An illustrative example based on the Canadian Consumer Price Index is presented.     

A Nonparametric Test Procedure Based on a Group of Quantile Tests

In this article, we consider nonparametric test procedures based on a group of quantile test statistics. We consider the quadratic form for the two-sided test and the maximal and summing types of statistics for the one-sided alternatives. Then we derive the null limiting distributions of the proposed test statistics using the large sample approximation theory. Also, we consider applying the permutation principle to obtain the null distribution. In this vein, we may consider the supremum type, which should use the permutation principle for obtaining the null distribution. Then we illustrate our procedure with an example and compare the proposed tests with other existing tests including the individual quantile tests by obtaining empirical powers through simulation study. Also, we comment on the related discussions to this testing procedure as concluding remarks. Finally we prove the lemmas and theorems in the appendices.     

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.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

Detecting Variance Change-Points for Blocked Time Series and Dependent Panel Data

This article proposes a class of weighted differences of averages (WDA) statistics to test and estimate possible change-points in variance for time series with weakly dependent blocks and dependent panel data without specific distributional assumptions. We derive the asymptotic distributions of the test statistics for testing the existence of a single variance change-point under the null and local alternatives. We also study the consistency of the change-point estimator. Within the proposed class of the WDA test statistics, a standardized WDA test is shown to have the best consistency rate and is recommended for practical use. An iterative binary searching procedure is suggested for estimating the locations of possible multiple change-points in variance, whose consistency is also established. Simulation studies are conducted to compare detection power and number of wrong rejections of the proposed procedure to that of a cumulative sum (CUSUM) based test and a likelihood ratio-based test. Finally, we apply the proposed method to a stock index dataset and an unemployment rate dataset. Supplementary materials for this article are available online.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

Divergences and duality for estimation and test under moment condition models

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alternatives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.     

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.     

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。     

Analysis of ordered covariate effects among groups with repeated measurements

Testing procedures for ordered covariate effects are developed in the repeated measures experiment. The maximum likelihood estimators of covariate effects under the ordered hypothesis are approximated by the isotonic regression of their unconstrained estimators. The asymptotic null distributions of the test statistics are chi-bar-square distributions which are mixtures of chi-square distributions. A Monte-Carlo simulation reveals that the proposed test for ordered covariate effects is seriously more powerful than the usual chi-square test that neglects the information on the order restriction. These testing methods are applied for analyzing the effect of vitamin E diet supplement on growth rate of animals.