Analysis of Means Using Ranks

A nonparametric procedure, called the analysis of means using ranks (ANOMR), is proposed for testing the equality of several population means. The ANOMR procedure may be used graphically in the form of a Shewhart control chart and so has the advantage of pinpointing which population mean, if any, is significantly different from the others. Exact and asymptotic critical values are given for the implementation of ANOMR. Results from a Monte Carlo power study are presented which indicate that for light-tailed distributions such as the uniform and the normal, ANOMR is only slightly less powerful than the parametric competitive procedures based on analysis of variance and analysis of means. For heavy-tailed distributions such as the Cauchy, ANOMR is shown to provide greater power than the parametric procedures. The results also indicate that for both light and heavy-tailed distributions the use of the ANOMR test instead of the Kruskal-Wallis test leads to only a small loss of power for a range of alternatives.     

Robust multivariate analysis of variability

A general testing procedure is proposed to multivariately test for equality of p variances among k groups. The procedure applies a multivariate analysis of variance on an appropriate measure of spread for the uncensored original observations. Three such measures of spread are compared in a simulation experiment which considered two and three variables with equal and unequal sample sizes for the null and alternative hypotheses for Gaussian, Student's t (8, 12, and 20 degrees of freedom) and gamma (α=2,4,6 and 10) distributions . The likelihood ratio test (Box, 1949) was included in the above simulations. The results suggest that if one chooses a measure of spread appropriate for the distribution of the original observations, the proposed MANOVA-based testing procedure is robust and reasonably powerful. Using this procedure for the normal distribution, similar power was observed to that of the likelihood ratio test when the variables were uncorrelated or had little positive correlation.     

Nonparametric tests for stratified additive hazards model based on current status data

Stratified regression models are commonly employed when study subjects may come from possibly different strata such as different medical centers, and for the situation, one common question of interest is to test the existence of the stratum effect. To address this, there exists some literature on the testing of the stratum effects under the framework of the proportional hazards model when one observes right-censored data or interval-censored data. In this paper, we consider the situation under the additive hazards model when one faces current status data, for which there does not seem to exist an established test procedure. The asymptotic distributions of the proposed test procedure are provided. Also a simulation study is performed to evaluate the performance of the proposed method and indicates that it works well for practical situations. The approach is applied to a set of real current status data from a tumorigenicity study.     

A tow-stage procedure for testing homogeneity of means against ordered alternatives in analysis of variance with unequal variances

In this paper we present a two-stage sampling procedure for testing the equality of normal means against ordered alternatives in one-way analysis of variance with unequal unknown variances. A table of approximated percentiles needed for implementation is provided. Some Monte Carlo results for estimating the power of the proposed test statistic are presented.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Some tests for a poisson process and their powers with respect to trend in the intensity

Recently many authors have worked on Wei bull process in the area of modelling and analysis. Much less work is done in the area of testing of hypothesis. In this article, some tests for testing the Poisson process against a class of Wei bull process based on the conditional distribution of observations given the sufficient statistic, are proposed. The percentage points of the distributions of the proposed test statistics are simulated. The powers of the tests under alternatives are computed by Monte Carlo method. It is seen that the suggested tests perform well for decreasing intensities.     

F-tests for seasonal differencing with a break-point

The classification between stochastic trend stationarity and deterministic broken trend stationarity is important because incorrect inferences can follow if a stationary series with a broken trend is incorrectly classified as integrated. In this paper, we consider joint tests for regular and seasonal unit roots null hypothesis against broken trend stationarity alternatives where the location of the break is known or unknown. Based on the F-test proposed by Hasza and Fuller (1982, Ann. Statist. 10, 1209–1216), we develop testing procedures for distinguishing these two types of process. The asymptotic distributions of test statistics are derived as functions of Wiener processes. A response surface regression analysis directed to relating the finite sample distributions and the breaking position is studied. Simulation experiments suggest that the power of the test is reasonable. The testing procedure is illustrated by the Canadian consumer price index series.     

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.     

Some tests for uniformity of circular distributions powerful against multimodal alternatives

The author introduces new statistics suited for testing uniformity of circular distributions and powerful against multimodal alternatives. One of them has a simple expression in terms of the geometric mean of the sample of chord lengths. The others belong to a family indexed by a continuous parameter. The asymptotic distributions under the null hypothesis are derived. We compare the power of the new tests against Stephens's alternatives with those of Ajne, Watson, and Hermans‐Rasson's tests. Some of the new tests are the most powerful when the alternative has three or four modes. A heuristic justification of this feature is given. An application to the analysis of archaeological data is provided. The Canadian Journal of Statistics 38:80–96; 2010 © 2010 Statistical Society of Canada     

Testing hypotheses in truncated samples by means of divergence statistics

In this paper we consider the problem of testing hypotheses in parametric models, when only the first r (of n) ordered observations are known.Using divergence measures, a procedure to test statistical hypotheses is proposed, Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained.Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators for truncated samples are considered.Applications of these results in testing statistical hypotheses, on the basis of truncated data, are presented.The small sample behavior of the proposed test statistics is analyzed in particular cases.A comparative study of power values is carried out by computer simulation.     

Simultaneous inferences for ordered exponential location parameters under unbalanced data and heteroscedasticity of scale parameters

A test procedure for testing homogeneity of location parameters against simple ordered alternative is proposed for k(k ≥ 2) members of two parameter exponential distribution under unbalanced data and heteroscedasticity of the scale parameters. The relevant one-sided and two-sided simultaneous confidence intervals (SCIs) for all k(k ? 1)/2 ordered pairwise differences of location parameters are also proposed. Simulation-based study revealed that the proposed procedure is better than the recently proposed procedure in terms of power, coverage probability, and average volume of SCIs. The implementation of proposed procedure is demonstrated through real life data.     

Subgroup Analysis with Time‐to‐Event Data Under a Logistic‐Cox Mixture Model

Subgroup detection has received increasing attention recently in different fields such as clinical trials, public management and market segmentation analysis. In these fields, people often face time‐to‐event data, which are commonly subject to right censoring. This paper proposes a semiparametric Logistic‐Cox mixture model for subgroup analysis when the interested outcome is event time with right censoring. The proposed method mainly consists of a likelihood ratio‐based testing procedure for testing the existence of subgroups. The expectation–maximization iteration is applied to improve the testing power, and a model‐based bootstrap approach is developed to implement the testing procedure. When there exist subgroups, one can also use the proposed model to estimate the subgroup effect and construct predictive scores for the subgroup membership. The large sample properties of the proposed method are studied. The finite sample performance of the proposed method is assessed by simulation studies. A real data example is also provided for illustration.     

Local influence analysis for regression models with scale mixtures of skew-normal distributions

The robust estimation and the local influence analysis for linear regression models with scale mixtures of multivariate skew-normal distributions have been developed in this article. The main virtue of considering the linear regression model under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows an easy implementation of inference. Inspired by the expectation maximization algorithm, we have developed a local influence analysis based on the conditional expectation of the complete-data log-likelihood function, which is a measurement invariant under reparametrizations. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and with Cook's well-known approach it can be very difficult to obtain measures of the local influence. Some useful perturbation schemes are discussed. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, a real data set has been analyzed, illustrating the usefulness of the proposed methodology.     

A sequential discrimination procedure for two almost identically shaped wrapped distributions

The way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.     

Analysis of binary data for the balanced one-way classification

A test is proposed for testing the equality of proportions based on the data available from a one-way classification having t treatment conditions and n binary observations per treatment. The test statistic B is a constant multiple of the F-statistic which results when the analysis of variance procedure for the one-way classification is applied to the data and, hence, is computationally simple. The statistic B from this binary analysis of variance (BIANOVA) is distributed asymptotically as a chi-square random variable. The proposed test is uniformly more powerful than either the F-test indicated above or the Pearson chi-square test; however, the attained empirical level of significance is frequently higher than for either of these competitors and usually higher than the stated level of significance for smaller values of n (say n ≤ 20).     

Empirical likelihood tests for two‐sample problems via nonparametric density estimation

The authors study the problem of testing whether two populations have the same law by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. They obtain the asymptotic distribution of the test statistic and propose a bootstrap approximation to calibrate the test. A simulation study is carried out in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions.     

Using principal components to test normality of high-dimensional data

Many multivariate statistical procedures are based on the assumption of normality and different approaches have been proposed for testing this assumption. The vast majority of these tests, however, are exclusively designed for cases when the sample size n is larger than the dimension of the variable p, and the null distributions of their test statistics are usually derived under the asymptotic case when p is fixed and n increases. In this article, a test that utilizes principal components to test for nonnormality is proposed for cases when p/n → c. The power and size of the test are examined through Monte Carlo simulations, and it is argued that the test remains well behaved and consistent against most nonnormal distributions under this type of asymptotics.     

Parametric bootstrap goodness-of-fit testing for Wehrly–Johnson bivariate circular distributions

The Wehrly–Johnson family of bivariate circular distributions is by far the most general one currently available for modelling data on the torus. It allows complete freedom in the specification of the marginal circular densities as well as the binding circular density which regulates any dependence that might exist between them. We propose a parametric bootstrap approach for testing the goodness-of-fit of Wehrly–Johnson distributions when the forms of their marginal and binding densities are assumed known. The approach admits the use of any test for toroidal uniformity, and we consider versions of it incorporating three such tests. Simulation is used to illustrate the operating characteristics of the approach when the underlying distribution is assumed to be bivariate wrapped Cauchy. An analysis of wind direction data recorded at a Texan weather station illustrates the use of the proposed goodness-of-fit testing procedure.     

The Multivariate Randomized Complete Block Design: A Novel Permutation Solution in Case of Ordered Categorical Variables

In this article, multivariate extensions of the combination-based test statistics for the comparison of several treatments in the multivariate Randomized Complete Block designs are introduced in case of categorical response variables. Several tests for the multivariate Randomized Complete Block designs, including MANOVA procedure, are compared with the method proposed via a Monte Carlo simulation study. The method has also been applied to a real case study in the field of sensorial testing studies. Results suggest that in each experimental situation where normality of the supposed underlying continuous model is hard to justify and especially when errors have heavy-tailed distributions, the proposed nonparametric procedure can be considered as a valid solution.     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.