Robust tests for equality of two population means under the normal model

A robust procedure is developed for testing the equality of means in the two sample normal model. This is based on the weighted likelihood estimators of Basu et al. (1993). When the normal model is true the tests proposed have the same asymptotic power as the two sample Student's t-statistic in the equal variance case. However, when the normality assumptions are only approximately true the proposed tests can be substantially more powerful than the classical tests. In a Monte Carlo study for the equal variance case under various outlier models the proposed test using Hellinger distance based weighted likelihood estimator compared favorably with the classical test as well as the robust test proposed by Tiku (1980).     

Testing the homogeneity of proportions for clustered binary data without knowing the correlation structure

A robust generalized score test for comparing groups of cluster binary data is proposed. This novel test is asymptotically valid for practically any underlying correlation configurations including the situation when correlation coefficients vary within or between clusters. This structure generally undermines the validity of the typical large sample properties of the method of maximum likelihood. Simulations and real data analysis are used to demonstrate the merit of this parametric robust method. Results show that our test is superior to two recently proposed test statistics advocated by other researchers.     

On the computation of a robusl version of the optimal c(α) test

A robust test of a parameter while in the presence of nuisance parameters was proposed by Wang (1981). The test procedure is a robust extension of the optimal C(α) tests. A numerical method for computing the solution of the orthogonality condition that is required by the test procedure is provided. An example on the testing of normal scale while in the presence of outliers is worked out to illustrate the construction of the robust test.     

A new jackknife test for homogeneity of variances

A new jackknife test is proposed to test the equality of variances in several populations. The new test is based on jackknifing one group of observations at a time, instead of one observation in each group as recommended by Miller for a two sample case, and by Layard for several samples. The proposed test is examined, and compared with other tests, in terms of power and robustness with respect to a wide variety of non-normal distributions. It is found that the new test is robust and has reasonably high power for normal as well as for non-normal observations, irrespective of the sample size. Furthermore, the proposed test is certainly superior to all other tests considered here in small to moderate size samples, and is as good as or better than the other tests in large samples, irrespective of the distribution of sampling observations.     

A Flexible Nonparametric Test for Ordered and Umbrella Alternatives

A flexible and robust test for the ordered and umbrella alternatives is proposed in this paper. A two-step procedure is presented to make the proposed test be easily applicable. Type I error and power of the given approach are thoroughly investigated by extensive Monte Carlo studies.     

A robust test for the homogeneity of scales

Many robust tests for the equality of variances have been proposed recently. Brown and Forsythe (1974) and Layard (1973) review some of the well-known procedures and compare them by simulation methods. Brown and Forsythe’s alternative formulation of Levene’s test statistic is found to be quite robust under certain nonnormal distributions. The performance of the methods, however, suffers in the presence of heavy tailed distributions such as the Cauchy distribution.

In this paper, we propose and study a simple robust test. The results obtained from the Monte Carlo study compare favorably with those of the existing procedures.     

Simple Robust Tests for Autocorrelated Errors in Time Series Design Intervention Models

A simple, robust test for the autocorrelation parameter in an intervention time-series model (AB design) is proposed. It is analogous to the traditional tests and can easily be computed by using the freeware R. In the same way as traditional tests of autocorrelation are based on least squares (LS) fits of a linear model, our robust test is based on the highly efficient Wilcoxon fit of the linear model. We present the results of a Monte Carlo study which show that our robust test inherits the good efficiency properties of this Wilcoxon fit. Its empirical power is only slightly less than the empirical power of the least squares test over situations with normally distributed errors while it exhibited much more power over situations with error distributions having tails heavier than those of a normal distribution. It also showed robustness of validity over all null situations simulated. We also present the results of the application of our test to a real data set which illustrates the robustness of our test.     

JACKKNIFE TESTS FOR HETEROSCEDASTICITY IN THE GENERAL LINEAR MODEL

In this paper, tests based on the Jackknife technique are proposed to test for heteroscedasticity in the linear regression model when the errors are non-normal. These are the Jackknifed Goldfeld-Quandt (GQ), and jack-knife related variations of White (H), Lagrange multiplier (LM), Glejser (GL) and Bickel (B) tests. The power of the proposed tests is compared with that of GQ, H, LM, GL and B tests; and the robustness to the error distribution is analyzed under several heteroscedastic assumptions. The GQ test is by far the best test if the error distribution is close to normal, however, GQ test is not robust against non-normal errors. By applying the jackknife technique to the regression a more robust statistic (GQJRG) is produced but the cost is a loss in power. The GQJRG statistic generally is not M powerful as the Bickel (BlOLS) and Glejser (GLlOLS) statistics.     

A robust test for asymptotic independence of bivariate extremes

We show that the existing tests for asymptotic independence are sensitive to outliers. A robust test is proposed. The new test is made stable under contamination through a shrinkage scheme. Simulations show that the new test performs well in the presence of contaminated data while maintaining good properties when there is no contamination. An application to real data shows the added value of our new robust approach.     

Higher-order asymptotics under model misspecification

Most of the higher-order asymptotic results in statistical inference available in the literature assume model correctness. The aim of this paper is to develop higher-order results under model misspecification. The density functions to O(n^?3/2) of the robust score test statistic and the robust Wald test statistic are derived under the null hypothesis, for the scalar as well as the multiparameter case. Alternate statistics which are robust to O(n^?3/2) are also proposed.     

A SIMPLE SEQUENTIAL WILCOXON TEST

A simple sequential non-parametric test for the two-sample problem is proposed. A method of deriving its O.C. and A.S.N. funtions is given, and their adequacy confirmed by simulation. The test is found to require about 10 percent more observations than an optimal rank test; however it is much easier to apply. The test is found to be relatively robust.     

A Novel Powerful Joint Analysis with Data Fusion in Two-stage Case–Control Genome-wide Association Studies

In genome-wide association studies (GWASs) to detect the disease-associated genetic variants, two-stage design has received much attention because of its cost effectiveness and high efficiency. Under the framework of a two-stage design, it has been shown that joint analysis is more powerful than replication-based analysis. Several robust tests have been proposed for joint analysis to handle the problem of unknown genetic mode of inheritance. However, existing joint analysis of combining test statistics from both stages might suffer from a loss of efficiency if the combined test statistics are not sufficient or the weight of the statistic for each stage is not appropriate. In this article, we propose a new strategy for joint analysis by combining the raw data rather than the test statistics across stages and construct a robust MAX3-based test for two-staged GWASs, which can make full use of the information of the data from both stages. Our numerical results show that the proposed procedure is more powerful and computationally much faster than the existing joint analysis procedures. An application to a type 2 diabetes dataset is used to illustrate the proposed approach.     

Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances

The problem of comparing several samples to decide whether the means and/or variances are significantly different is considered. It is shown that with very non-normal distributions even a very robust test to compare the means has poor properties when the distributions have different variances, and therefore a new testing scheme is proposed. This starts by using an exact randomization test for any significant difference (in means or variances) between the samples. If a non-significant result is obtained then testing stops. Otherwise, an approximate randomization test for mean differences (but allowing for variance differences) is carried out, together with a bootstrap procedure to assess whether this test is reliable. A randomization version of Levene's test is also carried out for differences in variation between samples. The five possible conclusions are then that (i) there is no evidence of any differences, (ii) evidence for mean differences only, (iii) evidence for variance differences only, (iv) evidence for mean and variance differences, or (v) evidence for some indeterminate differences. A simulation experiment to assess the properties of the proposed scheme is described. From this it is concluded that the scheme is useful as a robust, conservative method for comparing samples in cases where they may be from very non-normal distributions.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Local rank estimation and related test for varying-coefficient partially linear models

This paper develops a robust estimation procedure for the varying-coefficient partially linear model via local rank technique. The new procedure provides a highly efficient and robust alternative to the local linear least-squares method. In other words, the proposed method is highly efficient across a wide class of non-normal error distributions and it only loses a small amount of efficiency for normal error. Moreover, a test for the hypothesis of constancy for the nonparametric component is proposed. The test statistic is simple and thus the test procedure can be easily implemented. We conduct Monte Carlo simulation to examine the finite sample performance of the proposed procedures and apply them to analyse the environment data set. Both the theoretical and the numerical results demonstrate that the performance of our approach is at least comparable to those existing competitors.     

分布未知情况下的空间滞后模型检验

 在扰动项分布未知的情况下,直接采用传统的空间模型检验方法是存在问题的。针对传统空间模型检验方法的不足,本文以Lee和Yu（2010）的研究为基础,采用Lee和Liu（2006）提出的最优矩条件,构造分布未知情况下空间滞后模型的稳健检验统计量。这种检验方法仅需参数的一致估计量,便于计算。蒙特卡罗结果表明,在小样本情况下,本文提出的检验有良好的性质,且明显优于Saavedra（2003）提出的检验。     

Hypothesis Testing in Multivariate Linear Models with Randomly Missing Data

Tests for mean equality proposed by Weerahandi (1995) and Chen and Chen (1998), tests that do not require equality of population variances, were examined when data were not only heterogeneous but, as well, nonnormal in unbalanced completely randomized designs. Furthermore, these tests were compared to a test examined by Lix and Keselman (1998), a test that uses a heteroscedastic statistic (i.e., Welch, 1951) with robust estimators (20% trimmed means and Winsorized variances). Our findings confirmed previously published data that the tests are indeed robust to variance heterogeneity when the data are obtained from normal populations. However, the Weerahandi (1995) and Chen and Chen (1998) tests were not found to be robust when data were obtained from nonnormal populations. Indeed, rates of Type I error were typically in excess of 10% and, at times, exceeded 50%. On the other hand, the statistic presented by Lix and Keselman (1998) was generally robust to variance heterogeneity and nonnormality.     

A Robust Test for Non-nested Hypotheses

We propose a robust version of Cox-type test statistics for the choice between two non-nested hypotheses. We first show that the influence of small amounts of contamination in the data on the test decision can be very large. Secondly, we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable. Finally, we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution, unlike for the classical test statistic. We apply our results to the choice between a Pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept.     

Robust Testing for Skewness

Statistical analysis frequently involves the problem of assessing distributional properties. This article concerns the problem of testing for skewness of random variables. It is argued that the classical skewness test is not very useful for this purpose, and another approach is suggested that is easy to implement and is also robust to heteroscedasticity. The size, power, and robustness properties of the proposed test is evaluated and compared to the classical skewness test by means of Monte Carlo simulations.     

Approximate transformation trimmed mean methods to the test of simple linear regression slope equality

To deal with the problem of non-normality and heteroscedasticity, the current study proposes applying approximate transformation trimmed mean methods to the test of simple linear regression slope equality. The distribution-free slope estimates are first trimmed on both sides and then the test statistic t is transformed by Johnson's method for each group to correct non-normality. Lastly, an approximate test such as the James second-order test, the Welch test, or the DeShon-Alexander test, which are robust for heterogeneous variances, is applied to test the equality of regression slopes. Bootstrap methods and Monte Carlo simulation results show that the proposed methods provide protection against both unusual y values, as well as unusual x values. The new methods are valid alternatives for testing the simple linear regression slopes when heteroscedastic variances and nonnormality are present.