Generalized orthogonal contrast tests for homogeneity of ordered means

Several procedures have been proposed for testing equality of ordered means. The best-known of these is the likelihood-ratio test introduced by Bartholomew, which possesses generally superior power characteristics to those of its competitors. Difficulties in implementing this test have led to the development of alternative approaches, such as tests based on single and multiple contrasts. Some recent approaches have utilized approximations to the polyhedral cone defining the restricted parameter space, including those of Akkerboom (circular cone) and Mudholkar & McDermott (orthant). This article proposes a class of tests based on an improved orthant approximation to the polyhedral cone. These tests may be viewed as generalizations of the orthogonal contrast test proposed by Mukerjee, Robertson & Wright. Studies of the power functions of several competing tests indicate that the generalized orthogonal contrast tests are effective alternatives to the likelihood-ratio test, especially when the latter is difficult to implement.     

ROBUST TESTS FOR THE SIGNIFICANCE OF ORTHANT RESTRICTED MEAN VECTOR

Mudholkar and Srivastava [1] Mudholkar, G. S. and Srivastava, D. K. A class of robust stepwise alternatives to Hotelling's T²tests. Submitted to the Journal of Applied Statistics 1999 [Google Scholar]adapted Mudholkar and Subbaiah's [2] Mudholkar, G. S. and Subbaiah, P. 1980. Testing significance of a mean vector–a possible alternative to Hotelling's T². Ann. Inst. Statist. Math., 32(A): 43–52.  [Google Scholar]modified stepwise procedure, using the trimmed means in place of the means and appropriate studentization, to construct robust tests for the significance of a mean vector. They concluded that the robust alternatives provide excellent type I error control, and a substantial gain in power over Hotelling's T ²test in case of heavy tailed populations without significant loss of power when the population is normal. In this paper we adapt the modified stepwise approach to construct simple tests for the significance of the orthant constrained mean vector of a p-variate normal population with unknown covariance matrix, and also for constructing robust tests without assuming normality. The simple normal theory tests have exact type I error, whereas the robust tests provide a reasonably type I error control and substantial power advantage over Perlman's [3] Perlman, M. D. 1969. One-sided testing problems in multivariate analysis. Annals of Mathematical Statistics, 40: 549–567. [Crossref] , [Google Scholar]likelihood ratio test.     

The finite-sample performance of robust unit root tests

This paper investigates the relative small sample performance of several robust unit root tests by means of a simulation study. It is confirmed that the traditional least-squares based Dickey-Fuller test has substantially lower power than several robust alternatives if the error distribution is fat-tailed while its power gain is small at the normal model. Particularly good results are achieved by a quasi-maximum likelihood test. However, all robust tests under consideration exhibit severe size distortions if the disturbances follow a skewed distribution. Moreover, under additive outliers, robust tests fail to produce stable sizes and good power properties. Consequently, the value of using robust unit root tests depends heavily of the type of nonnormality at hand.     

A non-parametric test for umbrella alternatives based on ranked-set sampling

A test is proposed that extends the Chen-Wolfe (1990) test for umbrella alternatives with an unknown peak to use with ranked-set samples data. This follows from ideas in Bohn & Wolfe (1992), Magel (1994), and Hartlaub & Wolfe (1999). Critical values are simulated for the proposed test based on ranked-set samples of size 2 for 3, 4 and 5 populations. A power study is conducted comparing the proposed test using ranked-set samples with the Chen-Wolfe and Mack-Wolfe tests using simple random samples. Results are given.     

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).     

An empirical power study of multi-sample tests for exponentiality

Results are given of an empirical power study of three statistical procedures for testing for exponentiality of several independent samples. The test procedures are the Tiku (1974) test, a multi-sample Durbin (1975) test, and a multi-sample Shapiro–Wilk (1972) test. The alternative distributions considered in the study were selected from the gamma, Weibull, Lomax, lognormal, inverse Gaussian, and Burr families of positively skewed distributions. The general behavior of the conditional mean exceedance function is used to classify each alternative distribution. It is shown that Tiku's test generally exhibits overall greater power than either of the other two test procedures. For certain alternative distributions, Shapiro–Wilk's test is superior when the sample sizes are small.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.     

Testing the equality of location parameters of exponential populations from censored samples

The modified Tiku test and the modified likelihood ratio test proposed by Tiku and Vaughan (1991) for k=2 exponential populations are extended to . k > 2 populations. These tests are shown to be more powerful than the test proposed by Kambo and Awad (1985). Unlike the Kambo-Awad test, the proposed tests are shown to have almost symmetric power functions. Further, these tests can be applied when there is both left or right censoring present, in contrast to the tests of Sukhatme (1937), Bain and Englehardt (1991) and Elewa et al. (1992), who assume that there is no left censoring.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Testing equality of population variances the robust way

The purpose of this paper is to investigate the robustness (stability of Type I error to deviations from normality) and power properties of various tests for testing equality of population variances. It is shown that the tests based on Tiku’ s (1967, 1980, 1982) MML estimators have good robustness properties and are the most powerful overall.     

A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.     

Comparison of testing procedures utilizing P-values and Bayes factors in some common situations

Bayesian alternatives to classical tests for several testing problems are considered. One-sided and two-sided sets of hypotheses are tested concerning an exponential parameter, a Binomial proportion, and a normal mean. Hierarchical Bayes and noninformative Bayes procedures are compared with the appropriate classical procedure, either the uniformly most powerful test or the likelihood ratio test, in the different situations. The hierarchical prior employed is the conjugate prior at the first stage with the mean being the test parameter and a noninformative prior at the second stage for the hyper parameter(s) of the first stage prior. Fair comparisons are attempted in which fair means the likelihood of making a type I error is approximately the same for the different testing procedures; once this condition is satisfied, the power of the different tests are compared, the larger the power, the better the test. This comparison is difficult in the two-sided case due to the unsurprising discrepancy between Bayesian and classical measures of evidence that have been discussed for years. The hierarchical Bayes tests appear to compete well with the typical classical test in the one-sided cases.     

Tests for a simple tree order restriction with application to dose–response studies

Summary.  We propose 'Dunnett-type' test procedures to test for simple tree order restrictions on the means of p independent normal populations. The new tests are based on the estimation procedures that were introduced by Hwang and Peddada and later by Dunbar, Conaway and Peddada. The procedures proposed are also extended to test for 'two-sided' simple tree order restrictions. For non-normal data, nonparametric versions based on ranked data are also suggested. Using computer simulations, we compare the proposed test procedures with some existing test procedures in terms of size and power. Our simulation study suggests that the procedures compete well with the existing procedures for both one-sided and two-sided simple tree alternatives. In some instances, especially in the case of two-sided alternatives or for non-normally distributed data, the gains in power due to the procedures proposed can be substantial.     

Exact unconditional testing procedures for comparing two independent Poisson rates

We consider seven exact unconditional testing procedures for comparing adjusted incidence rates between two groups from a Poisson process. Exact tests are always preferable due to the guarantee of test size in small to medium sample settings. Han [Comparing two independent incidence rates using conditional and unconditional exact tests. Pharm Stat. 2008;7(3):195–201] compared the performance of partial maximization p-values based on the Wald test statistic, the likelihood ratio test statistic, the score test statistic, and the conditional p-value. These four testing procedures do not perform consistently, as the results depend on the choice of test statistics for general alternatives. We consider the approach based on estimation and partial maximization, and compare these to the ones studied by Han (2008) for testing superiority. The procedures are compared with regard to the actual type I error rate and power under various conditions. An example from a biomedical research study is provided to illustrate the testing procedures. The approach based on partial maximization using the score test is recommended due to the comparable performance and computational advantage in large sample settings. Additionally, the approach based on estimation and partial maximization performs consistently for all the three test statistics, and is also recommended for use in practice.     

Robust statistics for testing equality of means or variances

For testing the equality of means (location parameters) of two populations, Tiku (1980a) defined a statistic Tc (based on symmetrically censored samples) and showed that this statistic is robust to underlying populations and is also remarkably powerful. In this paper, we define a similar statistic T (based on samples s with observations censored only on one side) and show that this stat is tic is more powerful than T and nonparametric statistics, C for skew populations. We also provide a modification of this statistic for testing the equality of two population variances.     

Comparing J independent groups with a method based on trimmed means

The ANOVA-F test is the most popular and commonly used procedure for comparing J independent groups. However, it is well known that this method is very sensitive to non-normality, which has led to the derivation of alternative techniques based on robust estimators. In this work, ANOVA-F-test, trimmed mean Welch test, bootstrap-t trimmed mean Welch test, Schrader and Hettmansperger method with trimmed means, a percentile bootstrap method with trimmed means and a newly proposed method were compared in terms of both the Type I error probability and power. The proposed method compares well with ANOVA-F and other alternatives under various situations.     

STEPWISE TESTS WHEN THE TEST STATISTICS ARE INDEPENDENT

Consider the problem of simultaneously testing a nonhierarchical finite family of hypotheses based on independent test statistics. A general stepwise test is defined, of which the well known step-down and step-up tests are special cases. The step-up test is shown to dominate the other stepwise tests, including the step-down test, for situations of practical importance. When testing against two-sided alternatives, it is pointed out that if the step-up test is augmented to include directional decisions then the augmented step-up test controls the type I and III familywise error jointly at the original level q. The definition of the adjusted p values for the step-up test is justified. The results are illustrated by a numerical example.     

Goodness-of-fit tests for lifetime distributions based on Type II censored data

In this article, the general test statistic introduced by Alizadeh Noughabi and Balakrishnan [Goodness of fit using a new estimate of Kullback-Leibler information based on Type II censored data. IEEE Trans Reliab. 2015;64:627–635.] is applied for testing goodness of fit of lifetime distributions based on Type II censored data. The test statistic is constructed based on an estimate of Kullback–Leibler (KL) information. We investigate the properties of the proposed test statistic such as the test statistic is nonnegative, just like KL information. We apply this test statistic to following distributions: Exponential, Weibull, Log-normal and Pareto. The critical values and Type I error of the proposed tests are obtained. It is shown that the proposed tests have an excellent Type I error and hence can be used confidently in practice. Then, by Monte Carlo simulations, the power values of the proposed tests are computed against several alternatives and compared with those of the existing tests. Finally, some real-world reliability data are used for illustrative purpose.     

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.