High dimensional asymptotics for the naive Hotelling T2 statistic in pattern recognition

Abstract

This paper examines the high dimensional asymptotics of the naive Hotelling T² statistic. Naive Bayes has been utilized in high dimensional pattern recognition as a method to avoid singularities in the estimated covariance matrix. The naive Hotelling T² statistic, which is equivalent to the estimator of the naive canonical correlation, is a statistically important quantity in naive Bayes and its high dimensional behavior has been studied under several conditions. In this paper, asymptotic normality of the naive Hotelling T² statistic under a high dimension low sample size setting is developed using the central limit theorem of a martingale difference sequence.     

Regional efficacy evaluation in multi-regional clinical trials using a discounting factor weighted Z test

Multi-regional clinical trial (MRCT) is an efficient design to accelerate drug approval globally. Once the global efficacy of test drug is demonstrated, each local regulatory agency is required to prove effectiveness of test drug in their own population. Meanwhile, the ICH E5/E17 guideline recommends using data from other regions to help evaluate regional drug efficacy. However, one of the most challenges is how to manage to bridge data among multiple regions in an MRCT since various intrinsic and extrinsic factors exist among the participating regions. Furthermore, it is critical for a local agency to determine the proportion of information borrowing from other regions given the ethnic differences between target region and non-target regions. To address these issues, we propose a discounting factor weighted Z statistic to adaptively borrow information from non-target regions. In this weighted Z statistic, the weight is derived from a discounting factor in which the discounting factor denotes the proportion of information borrowing from non-target regions. We consider three ways to construct discounting factors based on the degree of congruency between target and non-target regions either using control group data, or treatment group data, or all data. We use the calibrated power prior to construct discounting factor based on scaled Kolmogorov–Smirnov statistic. Comprehensive simulation studies show that our method has desirable operating characteristics. Two examples are used to illustrate the applications of our proposed approach.     

最低工资标准与和谐劳动关系的构建——广州、佛山、深圳最低工资标准比较研究

最低工资保障问题是构建和谐劳动关系的核心内容和难点问题。对广州、佛山和深圳最低工资标准现状进行分析,并运用ELES模型测算其最低工资标准,测算结果显示：目前三市最低工资实际标准与理论标准仍有差距,其中广州的差距最小,佛山次之,深圳的差距最大;最低工资标准与职工平均工资比例偏低,与人均GDP比例关系失调,深圳最低工资执行标准总体优于广州和佛山。鉴此,提出应以发展线水平为目标制定最低工资标准等建议。     

Some Theoretical and Simulation Results on the Frequency Domain Causality Test

Breitung and Candelon (2006 Breitung , J. , Candelon , B. ( 2006 ). Testing for short- and long-run causality: A frequency-domain approach . Journal of Econometrics 132 : 363 – 378 .[Crossref], [Web of Science ®] , [Google Scholar]) in Journal of Econometrics proposed a simple statistical testing procedure for the noncausality hypothesis at a given frequency. In their paper, however, they reported some theoretical results indicating that their test severely suffers from quite low power when the noncausality hypothesis is tested at a frequency close to 0 or pi. This paper examines whether or not these results indicate their procedure is useless at such frequencies.     

Union-Intersection testing for outliers in multivariate normal data

The union-intersection approach to multivariate test construction is used to develop an alternative to Wilks' likelihood ratio test statistic for testing for two or more outliers in multivariate normal data. It is shown that critical values of both statistics are poorly approximated by Bonferroni bounds. Simulated critical values are presented for both statistics for significance levels 1% and 5%, for sample sizes 10(5)30, 40, 50, 75 and 100 for 2, 3, 4 and 5 dimensions. A power comparison of the two tests in the slippage of the mean model for generating outliers indicates that the union-intersection test is the more powerful when the slippages are close to collinear. Although Wilks' test remains the preference for general use, the union-intersection test could be valuable when such special structure in the data is suspected.     

Testing for order among K populations: theory and examples

Testing for stochastic order among K populations is a common and important problem in statistical practice. It arises in the analysis of both planned experiments and observational studies. The authors develop a new nonparametric test for order among K populations that can accommodate any stochastic ordering. The test is based on a maximally selected chi‐bar‐square statistic. The authors find its limiting distribution and use simulations to derive critical values. Three important examples are used to illustrate the applicability of the general method. The authors find that the new tests outperform the existing methods in many practical cases. The Canadian Journal of Statistics 38: 97–115; 2010 © 2009 Statistical Society of Canada     

An Optimal Semiparametric Method for Two‐group Classification

In the classical discriminant analysis, when two multivariate normal distributions with equal variance–covariance matrices are assumed for two groups, the classical linear discriminant function is optimal with respect to maximizing the standardized difference between the means of two groups. However, for a typical case‐control study, the distributional assumption for the case group often needs to be relaxed in practice. Komori et al. (Generalized t ‐statistic for two‐group classification. Biometrics 2015, 71: 404–416) proposed the generalized t ‐statistic to obtain a linear discriminant function, which allows for heterogeneity of case group. Their procedure has an optimality property in the class of consideration. We perform a further study of the problem and show that additional improvement is achievable. The approach we propose does not require a parametric distributional assumption on the case group. We further show that the new estimator is efficient, in that no further improvement is possible to construct the linear discriminant function more efficiently. We conduct simulation studies and real data examples to illustrate the finite sample performance and the gain that it produces in comparison with existing methods.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

Homogeneity test, sample size determination and interval construction of difference of two proportions in stratified bilateral-sample designs

In stratified otolaryngologic (or ophthalmologic) studies, the misleading results may be obtained when ignoring the confounding effect and the correlation between responses from two ears. Score statistic and Wald-type statistic are presented to test equality in a stratified bilateral-sample design, and their corresponding sample size formulae are given. Score statistic for testing homogeneity of difference between two proportions and score confidence interval of the common difference of two proportions in a stratified bilateral-sample design are derived. Empirical results show that (1) score statistic and Wald-type statistic based on dependence model assumption outperform other statistics in terms of the type I error rates; (2) score confidence interval demonstrates reasonably good coverage property; (3) sample size formula via Wald-type statistic under dependence model assumption is rather accurate. A real example is used to illustrate the proposed methodologies.     

Promoting tolerance for homosexuality: An evaluation of intervention strategies

This article presents a methodological critique of the few attempts to promote tolerance for homosexuality through education. The d statistic was utilized to assess the extent of attitude change from pre‐ to post‐intervention assessment. With few exceptions, data have been gathered from college students in courses focused on human sexuality. These data suggest that students who complete these courses become less fearful of contact with homosexuals, more tolerant of homosexuals in social roles, and more tolerant of homosexual behavior, although the extent and duration of the changes may be limited. Although some attribute attitude change to the intensiveness of the course, methodological and conceptual problems, including volunteer bias, provide alternative explanations. Further attempts to promote tolerance toward homosexuality must consider differences in attitudes toward gay men and lesbians and the extent to which men and women differ in their level of tolerance toward these groups.