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.     

Testing the equality of several means when the population variances are unequal

A comparative study is made of three tests, developed by James (1951), Welch (1951) and Brown & Forsythe (1974). James presented two methods of which only one is considered in this paper. It is shown that this method gives better control over the size than the other two tests. None of these methods is uniformly more powerful than the other two. In some cases the tests of James and Welch reject a false null hypothesis more often than the test of Brown & Forsythe, but there are also situations in which it is the other way around.

We conclude that for implementation in a statistical software package the very complicated test of James is the most attractive. A practical disadvantage of this method can be overcome by a minor modification.     

Testing equality of variances for several normal populations

In this study, we develop a test based on computational approach for the equality of variances of several normal populations. The proposed method is numerically compared with the existing methods. The numeric results demonstrate that the proposed method performs very well in terms of type I error rate and power of test. Furthermore we study the robustness of the tests by using simulation study when the underlying data are from t, exponential and uniform distributions. Finally we analyze a real dataset that motivated our study using the proposed test.     

Testing the equality of correlation matrices when sample correlation matrices are dependent

We consider the situation in which a 2m×2m$2m\times 2m$ correlation matrix is naturally partitioned into a 2×2$2\times 2$ form, where each submatrix is m×m$m\times m$ and we wish to test for the equality of the two correlation matrices appearing on the diagonal of the partitioned form. Standard tests for the equality of correlation matrices do not apply since the corresponding sample correlation matrices are correlated. We obtain Wald statistics under the assumption of multivariate normality as well as extensions that apply to elliptical distributions.     

Analysis of ranked data in randomized blocks when there are missing values

Data consisting of ranks within blocks are considered for randomized block designs when there are missing values. Tied ranks are possible. Such data can be analysed using the Skillings–Mack test. Here we suggest a new approach based on carrying out an ANOVA on the ranks using the general linear model platform available in many statistical packages. Such a platform allows an ANOVA to be calculated when there are missing values. Indicative sizes and powers show the ANOVA approach performs better than the Skillings–Mack test.     

Testing the equality means of several log-normal distributions

Researches propose various methods for comparing the means of two log-normal distributions. Some of these methods have been recently extended to test the equality means of several log-normal populations. Investigations show that none of the established methods is satisfactory. In this article, we provide three methods based on the computational approach test, which is a parametric bootstrap approach, for testing the means of several log-normal distributions. Further, we compare our methods with the existing methods through Monte Carlo simulation. The numerical results show that the Type I errors of these procedures are satisfactory regardless of the sample size, number of populations, and the true parameters. Finally, we explain the considered methods by real examples.     

Testing the equality of variance-covariance matrices the robust way

The robust statistic T² _Dproposed by Tiku and Singh (1982) for testing the equality of mean vectors of two mu1 t ivariate populations is modified to test the equality of variance-covariance matrices.     

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.     

Testing equality of correlation coefficients in two populations via permutation methods

The present paper investigates the asymptotic behaviour of a studentized permutation test for testing equality of (Pearson) correlation coefficients in two populations. It is shown that this test is asymptotically of exact level and has the same power for contiguous alternatives as the corresponding asymptotic test. As a by-product we specify the assumptions needed for the validity of the permutation test suggested in Sakaori (2002). A small simulation study compares the finite sample properties of the considered tests.     

Non-parametric estimation of lifetime distribution of competing risk models when censoring times are missing

There are situations in the analysis of failure time or lifetime data where the censoring times of unfailed units are missing. The non-parametric estimator of the lifetime distribution for such data is available in literature. In this paper we consider an extension of this situation to the univariate and bivariate competing risk setups. The maximum likelihood and simple moment estimators of cause specific distribution functions in both univariate and bivariate situations are developed. A simulation study is carried out to assess the performance of the estimators. Finally, we illustrate the method with real data set.     

On the distribution of the likelihood ratio test of equality of normal populations

It has recently been shown by Perlman (1980) that when testing the equality of several normal distributions it is the likelihood ratio test which is unbiased rather than a test based on a modified statistic in common use. This paper gives expansions for the null distribution of the likelihood ratio statistic as well as for the nonnull distribution in a special case.     

Testing for serial correlation in the presence of dynamic heteroscedasticity

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.     

Testing for serial correlation in the presence of dynamic heteroscedasticity

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.     

Estimation of the mean quality-adjusted survival using a multistate model for the sojourn times

In clinical trials, it may be of interest taking into account physical and emotional well-being in addition to survival when comparing treatments. Quality-adjusted survival time has the advantage of incorporating information about both survival time and quality-of-life. In this paper, we discuss the estimation of the expected value of the quality-adjusted survival, based on multistate models for the sojourn times in health states. Semiparametric and parametric (with exponential distribution) approaches are considered. A simulation study is presented to evaluate the performance of the proposed estimator and the jackknife resampling method is used to compute bias and variance of the estimator.     

A semi-Markov multistate model for estimation of the mean quality-adjusted survival for non-progressive processes

We discuss the estimation of the expected value of the quality-adjusted survival, based on multistate models. We generalize an earlier work, considering the sojourn times in health states are not identically distributed, for a given vector of covariates. Approaches based on semiparametric and parametric (exponential and Weibull distributions) methodologies are considered. A simulation study is conducted to evaluate the performance of the proposed estimator and the jackknife resampling method is used to estimate the variance of such estimator. An application to a real data set is also included.     

Estimating the association parameter for copula models under dependent censoring

Summary. Many biomedical studies involve the analysis of multiple events. The dependence between the times to these end points is often of scientific interest. We investigate a situation when one end point is subject to censoring by the other. The model assumptions of Day and co-workers and Fine and co-workers are extended to more general structures where the level of association may vary with time. Two types of estimating function are proposed. Asymptotic properties of the proposed estimators are derived. Their finite sample performance is studied via simulations. The inference procedures are applied to two real data sets for illustration.     

Assessing the fit of finite mixture distributions

Mixture distributions have become a very flexible and common class of distributions, used in many different applications, but hardly any literature can be found on tests for assessing their goodness of fit. We propose two types of smooth tests of goodness of fit for mixture distributions. The first test is a genuine smooth test, and the second test makes explicit use of the mixture structure. In a simulation study the tests are compared to some traditional goodness of fit tests that, however, are not customised for mixture distributions. The first smooth test has overall good power and generally outperforms the other tests. The second smooth test is particularly suitable for assessing the fit of each component distribution separately. The tests are applicable to both continuous and discrete distributions and they are illustrated on three medical data sets.     

An application of ranked set sampling when observations from several distributions are to be included in the sample

ABSTRACT

In this article, we propose a method to estimate the common location and common scale parameters of several distributions using suitably defined ranked set sampling. Efficiency comparison of the obtained estimators with some of the standard estimators is made. Illustration of the results to real life data sets is also described.     

Nonparametric estimation of survival functions for doubly censored observations when the lifetime hazard rate is proportionally related to the censoring hazard rates

Let T, X and Y be non-negative random variables, where T is the time of occurrence of an event of interest, X and Y being the lefl and right censoring variables respectively.

In this paper we propose a nonparametric estimator of the survival function, ST, when T, X and Y are supposed to be independent and their corresponding hazard rates are proportionally related. In this way, our results extend Ebrahimi's work (1985) to the doubly censored data case.     

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).