In this paper we explore the behaviour of dependent test statistics for testing of multiple hypothesis . To keep simplicity, we have considered a mixture normal model with equicorrelated correlation set up. With a simple linear transformation,the test statistics were decomposed into independent components, which, when conditioned appropriately generated independent variables. These were used to construct conditional tests , which were shown to be asymptotically optimal with power as large as that obtained using N.P.Lemma. We have pursued extensive simulation to support the claim.
Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the $\ell_p$-norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an e…
We investigate asymptotically optimal multiple testing procedures for streams of sequential data in the context of prior information on the number of false null hypotheses ("signals"). We show that the "gap" and "gap-intersection" procedures, recently proposed and shown by Song and Fellouris (2017, Electron. J. Statist.) to be asymptotically optimal for controlling type 1 and 2 familywise error rates (FWEs), are also asymptotically optimal for controlling FDR/FNR when their critical values are appropriately adjusted. Generalizing this result, we show that these procedures, again with appropria…
High-dimensional logistic regression is widely used in analyzing data with binary outcomes. In this paper, global testing and large-scale multiple testing for the regression coefficients are considered in both single- and two-regression settings. A test statistic for testing the global null hypothesis is constructed using a generalized low-dimensional projection for bias correction and its asymptotic null distribution is derived. A lower bound for the global testing is established, which shows that the proposed test is asymptotically minimax optimal over some sparsity range. For testing the in…
We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled ch…
The problem of multi-hypothesis testing with controlled sensing of observations is considered. The distribution of observations collected under each control is assumed to follow a single-parameter exponential family distribution. The goal is to design a policy to find the true hypothesis with minimum expected delay while ensuring that the probability of error is below a given constraint. The decision-maker can control the delay by intelligently choosing the control for observation collection in each time slot. We derive a policy that satisfies the given constraint on the error probability. We …
Stepwise multiple testing procedures have attracted several statisticians for decades and are also quite popular with statistics users because of their technical simplicity. The Bonferroni procedure has been one of the earliest and most prominent testing rules for controlling the familywise error rate (FWER). A recent article established that the FWER for the Bonferroni method asymptotically (i.e., when the number of hypotheses becomes arbitrarily large) approaches zero under any positively equicorrelated multivariate normal framework. However, similar results for the limiting behaviors of FWE…
We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test under the generalized Neyman-Pearson criterion. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated from anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests…
Asymptotically perfect seeded graph matching without edge correlation (and applications to inference)
Department of Mathematics, University of Maryland, College Park（数学系，马里兰大学，College Park）；University of Maryland, College Park（马里兰大学，College Park）；Human Language Technology Center of Excellence, Johns Hopkins University（语言技术卓越中心，约翰霍普金斯大学）
We present a general approach to constructing permutation tests that are both exact for the null hypothesis of equality of distributions and asymptotically correct for testing equality of parameters of distributions while allowing the distributions themselves to differ. These robust permutation tests transform a given test statistic by a consistent estimator of its limiting distribution function before enumerating its permutation distribution. This transformation, known as prepivoting, aligns the unconditional limiting distribution for the test statistic with the probability limit of its permu…
We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance…
Generalized linear models are often misspecified due to overdispersion, heteroscedasticity and ignored nuisance variables. Existing quasi-likelihood methods for testing in misspecified models often do not provide satisfactory type-I error rate control. We provide a novel semi-parametric test, based on sign-flipping individual score contributions. The tested parameter is allowed to be multi-dimensional and even high-dimensional. Our test is often robust against the mentioned forms of misspecification and provides better type-I error control than its competitors. When nuisance parameters are est…
High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local heritability. Group inference in regression models can be measured with respect to a weighted quadratic functional of the regression sub-vector corresponding to the group. Asymptotically unbiased estimators of these weighted quadratic functionals are constructed and a novel procedure using these estimators for inference is proposed. We derive its asymptotic Gaussia…
In the classical two-sample problem, the conventional approach for testing distributions equality is based on the difference between the two marginal empirical distribution functions, whereas a test for independence is based on the contrast between the bivariate and the product of the marginal empirical distribution functions. In this article we consider the problem of testing independence and distributions equality when the observer is "colour blind" so he cannot distinguish the distribution which has generated each of the two measurements. Within a nonparametric framework, we propose an empi…
We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) b…
For the mean vector test in high dimension, Ayyala et al.(2017,153:136-155) proposed new test statistics when the observational vectors are M dependent. Under certain conditions, the test statistics for one-same and two-sample cases were shown to be asymptotically normal. While the test statistics and the asymptotic results are valid, some parts of the proof of asymptotic normality need to be corrected. In this work, we provide corrections to the proofs of their main theorems. We also note a few minor discrepancies in calculations in the publication.
On large-sample estimation and testing via quadratic inference functions for correlated data
Hansen（1982）提出了一类“广义矩量法”（Gestivalized Method of Moments，Gestival），用于从一组评分函数中估计回归参数向量。Hansen证明了在一定的正则性条件下，基于Gynecology的估计量是相合的、渐近正态的和渐近有效的。在广义估计方程的框架下，扩展的原则，隐式地估计潜在的相关性结构，导致“二次推理函数”（QIF）的相关数据的分析。本研究的主要目的是：（1）为定义推理函数的扩展得分函数集建立一个合适的估计协方差阵：（2）为QIF建立一个统一的大样本理论框架;（三）推导出QIF检验统计量的一般化，用于涉及相关数据的一般线性假设问题，同时在零和局部备择假设;（4）提出了一种迭代加权广义最小二乘算法的推理QIF框架;（5）调查的基础矩阵的效果，定义了一组扩展的评分函数，通过Monte Carlo模拟实验的大小和功率的QIF测试。
We study asymptotic properties of Bayesian multiple testing procedures and provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian multiple testing procedure and associated error measures that coherently accounts for the dependence structure present in the model. We advocate posterior versions of FDR and FNR as appropriate error rates and show that their asymptotic convergence rates are directly associated with the Kullback-Leibler divergence from the true model. Our results hold even when the class of postulated models is miss…
In this article, we investigate the asymptotic properties of Bayesian multiple testing procedures under general dependent setup, when the sample size and the number of hypotheses both tend to infinity. Specifically, we investigate strong consistency of the procedures and asymptotic properties of different versions of false discovery and false non-discovery rates under the high dimensional setup. We particularly focus on a novel Bayesian non-marginal multiple testing procedure and its associated error rates in this regard. Our results show that the asymptotic convergence rates of the error rate…
The likelihood ratio statistic, with its asymptotic $χ^2$ distribution at regular model points, is often used for hypothesis testing. At model singularities and boundaries, however, the asymptotic distribution may not be $χ^2$, as highlighted by recent work of Drton. Indeed, poor behavior of a $χ^2$ for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood rat…
Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity
考虑独立正态观测均值的同时检验问题。在贝叶斯决策理论框架下，研究了一类单组收缩先验诱导的多重检验规则的渐近最优性，其中总损失为误分类假设的个数.我们假设数据为两组正态混合模型，并考虑Bogdan等人（2011）采用的渐近框架，他在多重检验的背景下引入了稀疏下渐近贝叶斯最优性的概念。一般类的一组先验研究是丰富的，包括，除其他外，家庭的三个参数β，广义双帕累托先验，特别是马蹄铁，正常指数伽玛和Strawderman-Berger先验。我们建立，在我们所选择的渐近框架内，研究中的多个测试规则渐近达到贝叶斯甲骨文的风险的乘法因子，与常数的风险接近的甲骨文风险的常数。这与Datta和Ghosh（2013）基于Carvalho等人（2009，2010）引入的马蹄估计量的多重检验规则获得的结果相似。我们进一步证明，在对底层稀疏参数的非常温和的假设下，基于van der Pas et al.（2014）提出的相应全局收缩参数的经验贝叶斯估计的诱导决策规则，在相同的乘法因子下渐近地获得最优贝叶斯风险。我们提供了一个统一的参数适用于一般类的先验研究。在此过程中，我们解决了Datta和Ghosh（2013）关于广义双重Pareto先验最优性的一个猜想。我们的工作还表明，Datta和Ghosh（2013）的结果可以进一步改进。
