Search

This paper is concerned with the problem of comparing the population means of two groups of independent observations. An approximate randomization test procedure based on the test statistic of Chen & Qin (2010) is proposed. The asymptotic behaviour of the test statistic, as well as the randomized statistic, is studied under weak conditions. In our theoretical framework, observations are not assumed to be identically distributed even within groups. No condition on the eigenstructure of the covariance matrices is imposed. Furthermore, the sample sizes of the two groups are allowed to be unbalanced. Under general conditions, all possible asymptotic distributions of the test statistic are obtained. We derive the asymptotic level and local power of the approximate randomization test procedure. Our theoretical results show that the proposed test procedure can adapt to all possible asymptotic distributions of the test statistic, always has the correct test level asymptotically and has good power behaviour. Our numerical experiments show that the proposed test procedure has favourable performance compared with several alternative test procedures. [ABSTRACT FROM AUTHOR]

There has been a growing interest in solving consensus optimization problems in a multi-agent system. It is known that there is an exactness-speed dilemma for decentralized optimization for the naive gradient method, where either we have fast linear convergence to a neighborhood of size O (η) using a fixed constant step size η , or slow convergence to the exact optimizer with a decreasing step size. We reinvestigate a special case of this problem from a statistical point of view, where each agent's local function is a sample estimate of a common population function. It is shown that when considering the convergence to the population target instead of to the exact consensual solution, the convergence is linear up to the statistical accuracy of the problem, using a step size only depending on the network characteristics instead of the sample size N. The objective function is not required to be strongly convex (but their statistical limit is), which allows for example the absolute deviation function which is used in median regression. Such results are in stark contrast to the optimization literature, where strong convexity of the objective function is necessary for linear convergence. [ABSTRACT FROM AUTHOR]

The Hosmer–Lemeshow test is a widely used method for evaluating the goodness of fit of logistic regression models. But its power is much influenced by the sample size, like other chi-square tests. Paul, Pennell, and Lemeshow (2013) considered using a large number of groups for large data sets to standardize the power. But simulations show that their method performs poorly for some models. In addition, it does not work when the sample size is larger than 25,000. In the present paper, we propose a modified Hosmer–Lemeshow test that is based on estimation and standardization of the distribution parameter of the Hosmer–Lemeshow statistic. We provide a mathematical derivation for obtaining the critical value and power of our test. Through simulations, we can see that our method satisfactorily standardizes the power of the Hosmer–Lemeshow test. It is especially recommendable for enough large data sets, as the power is rather stable. A bank marketing data set is also analyzed for comparison with existing methods. [ABSTRACT FROM AUTHOR]

The Wilcoxon–Mann–Whitney test is designed to test homogeneity of two random samples in the univariate case. It is very powerful to detect location shifts yet may lose power completely when there exist scale differences. We generalize the classic Wilcoxon–Mann–Whitney test through using pairwise distances of all observations. The generalized test can be readily used even when the random observations are multivariate. It is also very powerful in the presence of scale differences. The generalized test is in spirit to compare difference between the distribution functions of two random samples. It is m n / (m + n) -consistent under the strong null and local alternatives, and root- m n / (m + n) -consistent under fixed alternatives, where m , n stand for the respective sizes of the two random samples. The power of the generalized test is asymptotically independent of m / n , the size ratio of the two random samples. This indicates that the generalized test has nontrivial power as long as the sample sizes are not extremely unbalanced. We demonstrate the theoretical properties of our improved rank-based two-sample test through comprehensive numerical studies. [ABSTRACT FROM AUTHOR]

Testing for homogeneity of two random vectors is a fundamental problem in statistics. In the past two decades, numerous efforts have been made to detect heterogeneity when the random vectors are multivariate or even high dimensional. Due to the "curse of dimensionality", existing tests based on Euclidean distance may fail to capture the overall homogeneity in high-dimensional settings while can only capture the moment discrepancy. To address this issue, we propose a fully nonparametric test for homogeneity of two random vectors. Our method involves randomly selecting two subspaces consisting of components of the vectors, projecting the subspaces onto one-dimensional spaces, respectively, and constructing the test statistic using the Cramér–von Mises distance of the projections. To enhance the performance, we repeatedly implement this procedure to construct the final test statistic. Theoretically, if the replication time tends to infinity, we can avoid potential power loss caused by lousy directions. Owing to the U -statistic theory, the asymptotic null distribution of our proposed test is standard normal, regardless of the parent distributions of the random samples and the relationship between data dimensions and sample sizes. As a result, no re-sampling procedure is needed to determine critical values. The empirical size and power of the proposed test are demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]