There is growing concern about tacit collusion using algorithmic pricing, and regulators need tools to help detect the possibility of such collusion. This paper studies how to design a hypothesis testing framework in order to decide whether agents are behaving competitively or not. In our setting, agents are utility-maximizing and compete over prices of items. A regulator, with no knowledge of the agent's utility function, has access only to the agents' strategies (i.e., pricing decisions) and external shock values in order to decide if agents are behaving in competition according to some equilibrium problem. We leverage the formulation of such a problem as an inverse variational inequality and design a hypothesis test under a minimal set of assumptions. We demonstrate our method with computational experiments of the Bertrand competition game (with and without collusion) and show how our method performs.
In epidemiological studies, very often, evaluators obtain measurements of disease outcomes for study participants. In this paper, we propose a two-stage procedure for detecting outlier evaluators. In the first stage, a regression model is fitted to obtain the evaluators' effects. The outlier evaluators are considered as those with different effects compared with the normal evaluators. In the second stage, stepwise hypothesis testings are performed to detect outlier evaluators. The true positive rate and true negative rate of the proposed procedure are assessed in a simulation study. We apply t…
We formalize the problem of detecting the presence of a botnet in a network as an hypothesis testing problem where we observe a single instance of a graph. The null hypothesis, corresponding to the absence of a botnet, is modeled as a random geometric graph where every vertex is assigned a location on a $d$-dimensional torus and two vertices are connected when their distance is smaller than a certain threshold. The alternative hypothesis is similar, except that there is a small number of vertices, called the botnet, that ignore this geometric structure and simply connect randomly to every othe…
This paper proposes parametric and non-parametric hypothesis testing algorithms for detecting anisotropy -- rotational variance of the covariance function in random fields. Both algorithms are based on resampling mechanisms, which enable avoiding relying on asymptotic assumptions, as is common in previous algorithms. The algorithms' performance is illustrated numerically in simulation experiments and on real datasets representing a variety of potential challenges.
Epidemiologic and medical studies often rely on evaluators to obtain measurements of exposures or outcomes for study participants, and valid estimates of associations depends on the quality of data. Even though statistical methods have been proposed to adjust for measurement errors, they often rely on unverifiable assumptions and could lead to biased estimates if those assumptions are violated. Therefore, methods for detecting potential `outlier' evaluators are needed to improve data quality during data collection stage. In this paper, we propose a two-stage algorithm to detect `outlier' evalu…
We consider the "searching for a trail in a maze" composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side n based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero, mean off the anomalous path and mean μ_n on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. (2008) showed that if the unknown directed path under the signal hypothesis has known the initial location, then …
Consider a multiple hypothesis testing setting involving rare/weak effects: relatively few tests, out of possibly many, deviate from their null hypothesis behavior. Summarizing the significance of each test by a P-value, we construct a global test against the null using the Higher Criticism (HC) statistics of these P-values. We calibrate the rare/weak model using parameters controlling the asymptotic distribution of non-null P-values near zero. We derive a region in the parameter space where the HC test is asymptotically powerless. Our derivation involves very different tools than previously u…
This work constructs a hypothesis test for detecting whether an data-generating function $h: R^p \rightarrow R$ belongs to a specific reproducing kernel Hilbert space $\mathcal{H}_0$ , where the structure of $\mathcal{H}_0$ is only partially known. Utilizing the theory of reproducing kernels, we reduce this hypothesis to a simple one-sided score test for a scalar parameter, develop a testing procedure that is robust against the mis-specification of kernel functions, and also propose an ensemble-based estimator for the null model to guarantee test performance in small samples. To demonstrate th…
Meta-analysis techniques have been widely developed and applied in genomic applications, especially for combining multiple transcriptomic studies. In this paper we propose an order statistic of $p$-values ($r$th ordered $p$-value, rOP) across combined studies as the test statistic. We illustrate different hypothesis settings that detect gene markers differentially expressed (DE) 'in all studies," "in the majority of studies"' or "in one or more studies," and specify rOP as a suitable method for detecting DE genes "in the majority of studies." We develop methods to estimate the parameter $r$ in…
Indian Statistical Institute, Kolkata, India（印度统计学院）；University of California, Berkeley, USA（加州大学伯克利分校）；Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚高级研究院）；Barcelona School of Economics（巴塞罗那经济学院）；University of Leiden（莱顿大学）；Indian Statistical Institute, Delhi, India（印度统计学院德里中心）；Indian Institute of Technology Bombay（孟买印度理工学院）
This paper investigates the detection of information hidden by the Least Significant Bit (LSB) matching scheme. In a theoretical context of known image media parameters, two important results are presented. First, the use of hypothesis testing theory allows us to design the Most Powerful (MP) test. Second, a study of the MP test gives us the opportunity to analytically calculate its statistical performance in order to warrant a given probability of false-alarm. In practice when detecting LSB matching, the unknown image parameters have to be estimated. Based on the local estimator used in the W…
Detecting complex interactions among risk factors in case-control studies is a fundamental task in clinical and population research. However, though hypothesis testing using logistic regression (LR) is a convenient solution, the LR framework is poorly powered and ill-suited under several common circumstances in practice including missing or unmeasured risk factors, imperfectly correlated "surrogates", and multiple disease sub-types. The weakness of LR in these settings is related to the way in which the null hypothesis is defined. Here we propose the Asymmetric Independence Model (AIM) as a bi…
We consider the problem of recovering a subhypergraph based on an observed adjacency tensor corresponding to a uniform hypergraph. The uniform hypergraph is assumed to contain a subset of vertices called as subhypergraph. The edges restricted to the subhypergraph are assumed to follow a different probability distribution than other edges. We consider both weak recovery and exact recovery of the subhypergraph, and establish information-theoretic limits in each case. Specifically, we establish sharp conditions for the possibility of weakly or exactly recovering the subhypergraph from an informat…
The asymptotic distribution of the score test of the null hypothesis that marks do not impact the intensity of a Hawkes marked self-exciting point process is shown to be chi-squared. For local asymptotic power, the distribution against local alternatives is also established as non-central chi-squared. These asymptotic results are derived using existing asymptotic results for likelihood estimates of the unmarked Hawkes process model together with mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for t…
Detecting out-of-distribution (OOD) examples is critical in many applications. We propose an unsupervised method to detect OOD samples using a $k$-NN density estimate with respect to a classification model's intermediate activations on in-distribution samples. We leverage a recent insight about label smoothing, which we call the \emph{Label Smoothed Embedding Hypothesis}, and show that one of the implications is that the $k$-NN density estimator performs better as an OOD detection method both theoretically and empirically when the model is trained with label smoothing. Finally, we show that ou…
In spite of considerable practical importance, current algorithmic fairness literature lacks technical methods to account for underlying geographic dependency while evaluating or mitigating bias issues for spatial data. We initiate the study of bias in spatial applications in this paper, taking the first step towards formalizing this line of quantitative methods. Bias in spatial data applications often gets confounded by underlying spatial autocorrelation. We propose hypothesis testing methodology to detect the presence and strength of this effect, then account for it by using a spatial filter…
Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finitely many incorrect decisions as more data are collected. Said differently, our scheme almost certainly detects whether the process has a finite or infinite basis expansion for all sufficiently large sample sizes. Our app…
We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas.
In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős-Rényi graph with the same edge probability. If $n$ is the number of vertices and $α$ is the vector of eigenvalues, Eldan and Mikulincer show that detection is possible when $n^3 \gg (\|α\|_2/\|α\|_3)^6$ and impossible when $n^3 \ll (\|α\|_2/\|α\|_4)^4$. We show detection is impossible when $n^3 \ll…
Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control …
Rady School of Management（雷迪管理学院）；Department of Mathematics（数学系）；University of California at San Diego（加州大学圣地亚哥分校）
We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, i.e. model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging -- especially without making restrictive or unverifiable assumptions on the number of non-zero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inp…
We develop a group of robust, nonparametric hypothesis tests which detect differences between the covariance operators of several populations of functional data. These tests, called FKWC tests, are based on functional data depth ranks. These tests work well even when the data is heavy tailed, which is shown both in simulation and theoretically. These tests offer several other benefits, they have a simple distribution under the null hypothesis, they are computationally cheap and they possess transformation invariance properties. We show that under general alternative hypotheses these tests are …
Diagnosing the changes of structural behaviors using monitoring data is an important objective of structural health monitoring (SHM). The changes in structural behaviors are usually manifested as the feature changes in monitored structural responses; thus, developing effective methods for automatically detecting such changes is of considerable significance. Existing methods for change detection in SHM are mainly used for scalar or vector data, thus incapable of detecting the changes of the features represented by complex data, e.g., the probability density functions (PDFs). Detecting the abrup…
Bayesian hypothesis testing and minimax hypothesis testing represent extreme instances of detection in which the prior probabilities of the hypotheses are either completely and precisely known, or are completely unknown. Group minimax, also known as Gamma-minimax, is a robust intermediary between Bayesian and minimax hypothesis testing that allows for coarse or partial advance knowledge of the hypothesis priors by using information on sets in which the prior lies. Existing work on group minimax, however, does not consider the question of how to define the sets or groups of priors; it is assume…
We consider a setting in which we have a treatment and a large number of covariates for a set of observations, and wish to model their relationship with an outcome of interest. We propose a simple method for modeling interactions between the treatment and covariates. The idea is to modify the covariate in a simple way, and then fit a standard model using the modified covariates and no main effects. We show that coupled with an efficiency augmentation procedure, this method produces valid inferences in a variety of settings. It can be useful for personalized medicine: determining from a large s…
We consider the detection of activations over graphs under Gaussian noise, where signals are piece-wise constant over the graph. Despite the wide applicability of such a detection algorithm, there has been little success in the development of computationally feasible methods with proveable theoretical guarantees for general graph topologies. We cast this as a hypothesis testing problem, and first provide a universal necessary condition for asymptotic distinguishability of the null and alternative hypotheses. We then introduce the spanning tree wavelet basis over graphs, a localized basis that …
Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given α-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, w…
This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to spatial point processes. While the original embedding is infinite-dimensional and implicit, our approximate embedding is finite-dimensional and comes with explicit closed-form formulas. With its help we reduce the pattern comparison problem to the comparison of means in the Euclidean space. Hypothesis testing is based on conducting t-tests on each dimension…
A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null…
In this paper, we study the problem of testing the mean vectors of high dimensional data in both one-sample and two-sample cases. The proposed testing procedures employ maximum-type statistics and the parametric bootstrap techniques to compute the critical values. Different from the existing tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow general covariance structures of the data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step pr…
The repeated community-wide reuse of test sets in popular benchmark problems raises doubts about the credibility of reported test-error rates. Verifying whether a learned model is overfitted to a test set is challenging as independent test sets drawn from the same data distribution are usually unavailable, while other test sets may introduce a distribution shift. We propose a new hypothesis test that uses only the original test data to detect overfitting. It utilizes a new unbiased error estimate that is based on adversarial examples generated from the test data and importance weighting. Overf…
Large volume of networked streaming event data are becoming increasingly available in a wide variety of applications, such as social network analysis, Internet traffic monitoring and healthcare analytics. Streaming event data are discrete observation occurred in continuous time, and the precise time interval between two events carries a great deal of information about the dynamics of the underlying systems. How to promptly detect changes in these dynamic systems using these streaming event data? In this paper, we propose a novel change-point detection framework for multi-dimensional event data…
We consider the problem of testing for long-range dependence in time-varying coefficient regression models, where the covariates and errors are locally stationary, allowing complex temporal dynamics and heteroscedasticity. We develop KPSS, R/S, V/S, and K/S-type statistics based on the nonparametric residuals. Under the null hypothesis, the local alternatives as well as the fixed alternatives, we derive the limiting distributions of the test statistics. As the four types of test statistics could degenerate when the time-varying mean, variance, long-run variance of errors, covariates, and the i…
Interactions among people or objects are often dynamic in nature and can be represented as a sequence of networks, each providing a snapshot of the interactions over a brief period of time. An important task in analyzing such evolving networks is change-point detection, in which we both identify the times at which the large-scale pattern of interactions changes fundamentally and quantify how large and what kind of change occurred. Here, we formalize for the first time the network change-point detection problem within an online probabilistic learning framework and introduce a method that can re…
It has become an increasingly common practice for scientists in modern science and engineering to collect samples of multiple network data in which a network serves as a basic data object. The increasing prevalence of multiple network data calls for developments of models and theory that can deal with inference problems for populations of networks. In this work, we propose a general procedure for hypothesis testing of networks and in particular, for differentiating distributions of two samples of networks. We consider a very general framework which allows us to perform tests on large and spars…
The problem of detecting edge correlation between two Erdős-Rényi random graphs on $n$ unlabeled nodes can be formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are sampled independently; under the alternative, the two graphs are independently sub-sampled from a parent graph which is Erdős-Rényi $\mathbf{G}(n, p)$ (so that their marginal distributions are the same as the null). We establish a sharp information-theoretic threshold when $p = n^{-α+o(1)}$ for $α\in (0, 1]$ which sharpens a constant factor in a recent work by Wu, Xu and Yu. A key novelty in our w…
We study the two inference problems of detecting and recovering an isolated community of \emph{general} structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph $\mathcal{G}(n,q)$ with edge density $q\in(0,1)$; under the alternative, there is an unknown structure $Γ_k$ on $k$ nodes, planted in $\mathcal{G}(n,q)$, such that it appears as an \emph{induced subgraph}. In case of a successful detection, we are concerned with the task of recovering the correspon…
The statistical threshold for planted matchings and spanning trees
McGill University（麦吉尔大学）；University of British Columbia（不列颠哥伦比亚大学）；Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚高级研究院）；Barcelona Graduate School of Economics（巴塞罗那经济研究生院）；Northwestern University（西北大学）；Stanford University（斯坦福大学）
