Testing the equality of matrix distributions

Statistical Methods & Applications - While matrices are usually used as the basic data structure for experiments with repeated measurements or longitudinal data, testing methods for the...     

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 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 two intercepts for the parallel regression model with non sample prior information

This paper proposes tests for equality of intercepts of two simple regression models when non sample prior information is available on the equality of two slopes. For three different scenarios on the values of the slope, namely (i) unknown (unspecified), (ii) known (specified), and (iii) suspected, we derive the unrestricted test (UT), restricted test (RT), and pre-test test (PTT) for testing the equality of intercepts. The test statistics, their sampling distributions, and power functions of the tests are obtained. Comparison of power function and size of the tests reveals that the PTT has a reasonable dominance over the UT and RT.     

High breakdown point robust estimators with missing data

In this paper, we propose a new procedure to estimate the distribution of a variable y when there are missing data. To compensate the presence of missing responses, it is assumed that a covariate vector x is observed and that y and x are related by means of a semi-parametric regression model. Observed residuals are combined with predicted values to estimate the missing response distribution. Once the responses distribution is consistently estimated, we can estimate any parameter defined through a continuous functional T using a plug in procedure. We prove that the proposed estimators have high breakdown point.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

Bounded-leverage weights for robust regression estimators

Both the least squares estimator and M-estimators of regression coefficients are susceptible to distortion when high leverage points occur among the predictor variables in a multiple linear regression model. In this article a weighting scheme which enables one to bound the leverage values of a weighted matrix of predictor variables is proposed. Bounded-leverage weighting of the predictor variables followed by M-estimation of the regression coefficients is shown to be effective in protecting against distortion due to extreme predictor-variable values, extreme response values, or outlier-induced multieollinearites. Bounded-leverage estimators can also protect against distortion by small groups of high leverage points.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Equivariant estimators of the covariance matrix

Given a Wishart matrix S [S ∽ W_p(n, Σ)] and an independent multinomial vector X [X ∽ N_p (μ, Σ)], equivariant estimators of Σ are proposed. These estimators dominate the best multiple of S and the Stein-type truncated estimators.     

Testing the equality of several covariance matrices

Using a new approach based on Meijer G-functions and computer simulation, we numerically compute the exact null distribution of the modified-likelihood ratio statistic used to test the hypothesis that several covariances matrices of normal distributions are equal. Small samples of different sizes are considered, and for the case of two matrices, we introduce a new test based on determinants, with the null distribution of its criterion also fully computable. Comparisons with published results show the accuracy of our approach, which is proved to be more flexible and adaptable to different cases.     

Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

A comparison of doubly robust estimators of the mean with missing data

We consider data with a continuous outcome that is missing at random and a fully observed set of covariates. We compare by simulation a variety of doubly-robust (DR) estimators for estimating the mean of the outcome. An estimator is DR if it is consistent when either the regression model for the mean function or the propensity to respond is correctly specified. Performance of different methods is compared in terms of root mean squared error of the estimates and width and coverage of confidence intervals or posterior credibility intervals in repeated samples. Overall, the DR methods tended to yield better inference than the incorrect model when either the propensity or mean model is correctly specified, but were less successful for small sample sizes, where the asymptotic DR property is less consequential. Two methods tended to outperform the other DR methods: penalized spline of propensity prediction [Little RJA, An H. Robust likelihood-based analysis of multivariate data with missing values. Statist Sinica. 2004;14:949–968] and the robust method proposed in [Cao W, Tsiatis AA, Davidian M. Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika. 2009;96:723–734].     

Usefulness of robust estimators in sample survey

It Is illustrated that the adaptation of robust estimators, Tiku's (1967, 1978, 1980) modified maximum likelihood estimators for example, in sample survey can lead to considerable improvements In efficiencies in estimating the mean of a finite population (a random sample of Independent observations from a super-popularlon). In this paper, however, we confine our attention to symmetric super-populations     

A note on james type robust estimators

M-estimation of a single parameter of the life time distribution is considered based on independent and identically distributed survival data which may be randomly censored. The most robust and the optimal robust M-estimators of the location parameters of the survival time distribution are derived within a class considered in James (1986) as well as for the general unrestricted class. The properties of the estimators corresponding to the above two classes are discussed. A data set is used to illustrate the usefulness of the optimal robust estimators for the parameter of extreme value distribution.     

Optimally robust estimators in generalized Pareto models

In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location–dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér–von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the package evir.     

Preliminary estimators for robust non-linear regression estimation

In this paper, the robustness of weighted non-linear least-squares estimation based on some preliminary estimators is examined. The preliminary estimators are the Lnorm estimates proposed by Schlossmacher, by El-Attar et al., by Koenker and Park, and by Lawrence and Arthur. A numerical example is presented to compare the robustness of the weighted non-linear least-squares approach when based on the preliminary estimators of Schlossmacher (HS), El-Attar et al. (HEA), Koenker and Park (HKP), and Lawrence and Arthur (HLA). The study shows that the HEA estimator is as robust as the HKP estimator. However, the HEA estimator posed certain computational problems and required more storage and computing time.     

Modified ratio estimators using robust regression methods

When there is an outlier in the data set, the efficiency of traditional methods decreases. In order to solve this problem, Kadilar et al. (2007) adapted Huber-M method which is only one of robust regression methods to ratio-type estimators and decreased the effect of outlier problem. In this study, new ratio-type estimators are proposed by considering Tukey-M, Hampel M, Huber MM, LTS, LMS and LAD robust methods based on the Kadilar et al. (2007). Theoretically, we obtain the mean square error (MSE) for these estimators. We compared with MSE values of proposed estimators and MSE values of estimators based on Huber-M and OLS methods. As a result of these comparisons, we observed that our proposed estimators give more efficient results than both Huber M approach which was proposed by Kadilar et al. (2007) and OLS approach. Also, under all conditions, all of the other proposed estimators except Lad method are more efficient than robust estimators proposed by Kadilar et al. (2007). And, these theoretical results are supported with the aid of a numerical example and simulation by basing on data that includes an outlier.     

Testing equality of several exponential distributions

Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.     

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.