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) based tests attain the optimal type-II error exponent on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one- and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.
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…
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 present a general framework for hypothesis testing on distributions of sets of individual examples. Sets may represent many common data sources such as groups of observations in time series, collections of words in text or a batch of images of a given phenomenon. This observation pattern, however, differs from the common assumptions required for hypothesis testing: each set differs in size, may have differing levels of noise, and also may incorporate nuisance variability, irrelevant for the analysis of the phenomenon of interest; all features that bias test decisions if not accounted for. I…
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）
Knowing the error distribution is important in many multivariate time series applications. To alleviate the risk of error distribution mis-specification, testing methodologies are needed to detect whether the chosen error distribution is correct. However, the majority of the existing tests only deal with the multivariate normal distribution for some special multivariate time series models, and they thus can not be used to testing for the often observed heavy-tailed and skewed error distributions in applications. In this paper, we construct a new consistent test for general multivariate time se…
Otto-von-Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学）；Universidad de Valparaíso（瓦尔帕莱索大学）
In this paper we propose a Multiple kernel testing procedure to infer survival data when several factors (e.g. different treatment groups, gender, medical history) and their interaction are of interest simultaneously. Our method is able to deal with complex data and can be seen as an alternative to the omnipresent Cox model when assumptions such as proportionality cannot be justified. Our methodology combines well-known concepts from Survival Analysis, Machine Learning and Multiple Testing: differently weighted log-rank tests, kernel methods and multiple contrast tests. By that, complex hazard…
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…
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 …
A common challenge in nonparametric inference is its high computational complexity when data volume is large. In this paper, we develop computationally efficient nonparametric testing by employing a random projection strategy. In the specific kernel ridge regression setup, a simple distance-based test statistic is proposed. Notably, we derive the minimum number of random projections that is sufficient for achieving testing optimality in terms of the minimax rate. An adaptive testing procedure is further established without prior knowledge of regularity. One technical contribution is to establi…
Robust Kernel Hypothesis Testing under Data Corruption
Centre for Artificial Intelligence（人工智能中心）；Gatsby Computational Neuroscience Unit（加布特计算神经科学单位）；University College London（伦敦大学学院）；Inria London（伦敦研究所）；Department of Statistics and Data Science（统计与数据科学系）；Department of Applied Statistics（应用统计学系）；Yonsei University（延世大学）
Equivalence of distance-based and RKHS-based statistics in hypothesis testing
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Institute of Statistical Mathematics（统计数学研究所）
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…
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…
University College London（伦敦大学学院）；University of Technology Sydney（悉尼科技大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We propose a class of kernel-based two-sample tests, which aim to determine whether two sets of samples are drawn from the same distribution. Our tests are constructed from kernels parameterized by deep neural nets, trained to maximize test power. These tests adapt to variations in distribution smoothness and shape over space, and are especially suited to high dimensions and complex data. By contrast, the simpler kernels used in prior kernel testing work are spatially homogeneous, and adaptive only in lengthscale. We explain how this scheme includes popular classifier-based two-sample tests as…
Testing Hypotheses by Regularized Maximum Mean Discrepancy
IDSIA/SUPSI；Università della Svizzera Italiana（瑞士意大利语区大学）；Vita-Salute San Raffaele University（圣拉斐尔生命健康大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）；Université de Sherbrooke（舍布鲁克大学）
Google Research（谷歌研究院）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；University College London（伦敦大学学院）
We propose two nonparametric statistical tests of goodness of fit for conditional distributions: given a conditional probability density function $p(y|x)$ and a joint sample, decide whether the sample is drawn from $p(y|x)r_x(x)$ for some density $r_x$. Our tests, formulated with a Stein operator, can be applied to any differentiable conditional density model, and require no knowledge of the normalizing constant. We show that 1) our tests are consistent against any fixed alternative conditional model; 2) the statistics can be estimated easily, requiring no density estimation as an intermediate…
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…
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…
We study the kernel estimator of the transition density of bifurcating Markov chains. Under some ergodic and regularity properties, we prove that this estimator is consistent and asymptotically normal. Next, in the numerical studies, we propose two data-driven methods to choose the bandwidth parameters. These methods are based on the so-called two bandwidths approach.
In this article, we propose a class of $L_q$-norm based U-statistics for a family of global testing problems related to high-dimensional data. This includes testing of mean vector and its spatial sign, simultaneous testing of linear model coefficients, and testing of component-wise independence for high-dimensional observations, among others. Under the null hypothesis, we derive asymptotic normality and independence between $L_q$-norm based U-statistics for several $q$s under mild moment and cumulant conditions. A simple combination of two studentized $L_q$-based test statistics via their $p$-…
Inference for Multi-Dimensional High-Frequency Data: Equivalence of Methods, Central Limit Theorems, and an Application to Conditional Independence Testing
This paper aims to solve a basic problem in distributed statistical inference: how many machines can we use in parallel computing? In kernel ridge regression, we address this question in two important settings: nonparametric estimation and hypothesis testing. Specifically, we find a range for the number of machines under which optimal estimation/testing is achievable. The employed empirical processes method provides a unified framework, that allows us to handle various regression problems (such as thin-plate splines and nonparametric additive regression) under different settings (such as univa…
We propose a kernel machine based hypothesis testing procedure in nonlinear function-on-scalar regression model. Our research is motivated by the Newborn Epigenetic Study (NEST) where the question of interest is whether a pre-specified group of toxic metals or methylation at any of 9 differentially methylated regions (DMRs) is associated with child growth. We take the child growth trajectory as the functional response, and model the toxic metal measurements jointly using a nonlinear function. We use a kernel machine approach to model the unknown function and transform the hypothesis of no effe…
We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We show that when $n$ independent copies of a point process in $\mathbb R^d$ are superposed, the optimal bandwidth $h_n$ is of the order $n^{-1/(d+4)}$ under appropriate smoothness conditions on the kernel and true intensity function. We apply the Abramson principle to define adaptive kernel estimators and show that asymptotically the optimal adaptive bandwidth is of the order $n^{-1/(d+8)}$ under appropriate smoothness conditions.
