Composite Hypothesis Testing Problem Research Articles

The Geometry of Generalized Likelihood Ratio Test.

The generalized likelihood ratio test (GLRT) for composite hypothesis testing problems is studied from a geometric perspective. An information-geometrical interpretation of the GLRT is proposed based on the geometry of curved exponential families. Two geometric pictures of the GLRT are presented for the cases where unknown parameters are and are not the same under the null and alternative hypotheses, respectively. A demonstration of one-dimensional curved Gaussian distribution is introduced to elucidate the geometric realization of the GLRT. The asymptotic performance of the GLRT is discussed based on the proposed geometric representation of the GLRT. The study provides an alternative perspective for understanding the problems of statistical inference in the theoretical sense.

Multistatic Passive Radar Target Detection Under Uncalibrated Receivers With Direct-Path Interference

This article examines the multistatic passive radar target detection problem under uncalibrated receivers with direct-path interference. To this aim, we consider a passive multistatic radar consisting of several transmitters nonoverlapping in frequency with a common bandwidth and several spatially separated receivers without exploiting the reference channel. For uncalibrated, multichannel and multifrequency receivers and in the presence of direct-path interference signal, we model the target detection problem as a composite hypothesis testing problem and apply the Rao framework to obtain a closed-form test statistic. Monte Carlo simulation results are provided to illustrate the performance of the proposed Rao detector and compare it with several well-known passive radar target detection algorithms. The results show that the proposed detector significantly outperforms its counterparts for different multistatic configurations in terms of detection probability and false alarm regulation as well.

Finite-Sample Two-Group Composite Hypothesis Testing via Machine Learning

In the problem of composite hypothesis testing, identifying the potential uniformly most powerful (UMP) unbiased test is of great interest. Beyond typical hypothesis settings with exponential family, it is usually challenging to prove the existence and further construct such UMP unbiased tests with finite sample size. For example in the COVID-19 pandemic with limited previous assumptions on the treatment for investigation and the standard of care, adaptive clinical trials are appealing due to ethical considerations, and the ability to accommodate uncertainty while conducting the trial. Although several methods have been proposed to control Type I error rates, how to find a more powerful hypothesis testing strategy is still an open question. Motivated by this problem, we propose an automatic framework of constructing test statistics and corresponding critical values via machine learning methods to enhance power in a finite sample. In this article, we particularly illustrate the performance using Deep Neural Networks (DNN) and discuss its advantages. Simulations and two case studies of adaptive designs demonstrate that our method is automatic, general and prespecified to construct statistics with satisfactory power in finite-sample. Supplemental materials are available online including R code and an R shiny app.

Generalized Likelihood Ratio Test for Adversarially Robust Hypothesis Testing

Machine learning models are known to be susceptible to adversarial attacks which can cause misclassification by introducing small but well designed perturbations. In this paper, we consider a classical hypothesis testing problem in order to develop fundamental insight into defending against such adversarial perturbations. We interpret an adversarial perturbation as a nuisance parameter, and propose a defense based on applying the generalized likelihood ratio test (GLRT) to the resulting composite hypothesis testing problem, jointly estimating the class of interest and the adversarial perturbation. While the GLRT approach is applicable to general multi-class hypothesis testing, we first evaluate it for binary hypothesis testing in white Gaussian noise under $\ell_{\infty}$ norm-bounded adversarial perturbations, for which a known minimax defense optimizing for the worst-case attack provides a benchmark. We derive the worst-case attack for the GLRT defense, and show that its asymptotic performance (as the dimension of the data increases) approaches that of the minimax defense. For non-asymptotic regimes, we show via simulations that the GLRT defense is competitive with the minimax approach under the worst-case attack, while yielding a better robustness-accuracy tradeoff under weaker attacks. We also illustrate the GLRT approach for a multi-class hypothesis testing problem, for which a minimax strategy is not known, evaluating its performance under both noise-agnostic and noise-aware adversarial settings, by providing a method to find optimal noise-aware attacks, and heuristics to find noise-agnostic attacks that are close to optimal in the high SNR regime.

A comprehensive gene-centric pleiotropic association analysis for 14 psychiatric disorders with GWAS summary statistics.

BackgroundRecent genome-wide association studies (GWASs) have revealed the polygenic nature of psychiatric disorders and discovered a few of single-nucleotide polymorphisms (SNPs) associated with multiple psychiatric disorders. However, the extent and pattern of pleiotropy among distinct psychiatric disorders remain not completely clear.MethodsWe analyzed 14 psychiatric disorders using summary statistics available from the largest GWASs by far. We first applied the cross-trait linkage disequilibrium score regression (LDSC) to estimate genetic correlation between disorders. Then, we performed a gene-based pleiotropy analysis by first aggregating a set of SNP-level associations into a single gene-level association signal using MAGMA. From a methodological perspective, we viewed the identification of pleiotropic associations across the entire genome as a high-dimensional problem of composite null hypothesis testing and utilized a novel method called PLACO for pleiotropy mapping. We ultimately implemented functional analysis for identified pleiotropic genes and used Mendelian randomization for detecting causal association between these disorders.ResultsWe confirmed extensive genetic correlation among psychiatric disorders, based on which these disorders can be grouped into three diverse categories. We detected a large number of pleiotropic genes including 5884 associations and 2424 unique genes and found that differentially expressed pleiotropic genes were significantly enriched in pancreas, liver, heart, and brain, and that the biological process of these genes was remarkably enriched in regulating neurodevelopment, neurogenesis, and neuron differentiation, offering substantial evidence supporting the validity of identified pleiotropic loci. We further demonstrated that among all the identified pleiotropic genes there were 342 unique ones linked with 6353 drugs with drug-gene interaction which can be classified into distinct types including inhibitor, agonist, blocker, antagonist, and modulator. We also revealed causal associations among psychiatric disorders, indicating that genetic overlap and causality commonly drove the observed co-existence of these disorders.ConclusionsOur study is among the first large-scale effort to characterize gene-level pleiotropy among a greatly expanded set of psychiatric disorders and provides important insight into shared genetic etiology underlying these disorders. The findings would inform psychiatric nosology, identify potential neurobiological mechanisms predisposing to specific clinical presentations, and pave the way to effective drug targets for clinical treatment.

A Unified Framework for Multistatic Passive Radar Target Detection Under Uncalibrated Receivers

This work addresses the target-detection problem in a passive multistatic radar consisting of one transmitter and several spatially separated receivers (i.e., single-input multiple-output (SIMO) configuration) without exploiting the reference channel (RC). Several solutions for this problem are developed within the Rao, likelihood ratio test (LRT) and geometrical representation frameworks as well as under different assumptions on parameter space of this problem. Thus, a unified framework for multistatic passive radar target detection is created. To do this, we formulate three different target detection problems as composite hypothesis testing problems, where they differ based on different assumptions on the parameter space of the alternative hypothesis, resulting in four new detectors. We analytically prove that most of the proposed target detection methods posses constant false alarm rate (CFAR) behavior against the noise variance uncertainties across different receivers. Then, through Monte-Carlo simulations, we examine the detection performance of the proposed detectors to show the capabilities of the proposed detectors.

Exploiting Channel Gain and Phase Noise for PHY-Layer Authentication in Massive MIMO Systems

By exploiting two intrinsic physical (PHY)-layer features in terms of location-specific channel gain and transmitter-specific phase noise, this paper proposes a new PHY-layer authentication scheme for massive multiple-input multiple-output (MIMO) systems. In particular, we apply the linear minimum mean square error technique to estimate the time-varying channel gain and adopt extended Kalman filtering to track the time-varying phase noise. Based on the estimation error covariance matrices of channel gain and phase noise, we then formulate the PHY-layer authentication as a composite hypothesis testing problem. With the help of tools from statistical signal processing, matrix analysis, and composite hypothesis testing, we develop theoretical models to capture the false alarm and detection probability performances of the proposed scheme. Finally, we provide extensive numerical results to validate these theoretical models and to illustrate the efficiency of the proposed authentication scheme.

Multistatic Passive Radar Sensing Algorithms With Calibrated Receivers

This paper addresses a multistatic target sensing problem with a calibrated receiver with an unknown but deterministic ambient signal. To do this, we formulate two different target detection problems as composite hypothesis testing problems, where they differ based on whether the reference channel (RC) is available or not. Indeed, the detection problem 1 (P1) is formulated for the case in which each receiver utilizes an RC to collect the ambient signal and a surveillance channel (SC) for sensing potential target echo, while the second detection problem (P2) is for the case without RCs. Then, we resort to the likelihood ratio test (LRT) principle to devise two detection methods, namely P1-LRT and P2-LRT detectors. For the proposed P1-LRT method, we show that any uncertainty in the value of averaged direct signal-to-noise-ratio of the available reference channels, abbreviated as DNR <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">avg</sub> , can result in excessive false alarm probability in low DNR <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">avg</sub> regime. To facilitate efficient operation under uncertainty in DNR <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">avg</sub> , we propose a new threshold-setting strategy to adjust the level of the proposed P1-LRT detector. Through extensive Monte-Carlo (MC) simulations, we examine the above challenge and investigate the efficiency of the proposed threshold-setting strategy. Although the P1-LRT has a fixed level and the P2-LRT is a fixed size test, the P1-LRT detector significantly outperforms the P2-LRT, as indicated by the MC simulation results.

Decision fusion for composite hypothesis testing in wireless sensor networks over a shared and noisy collision channel

We consider the composite hypothesis testing problem of time-bandwidth-constrained distributed detection. In this scenario, the probability distribution of the observed signal when the event of interest is happening is unknown. In addition, local decisions are censored and only those uncensored local decisions will be sent to the fusion center over a shared and noisy collision channel. The fusion center also has a limited time duration to collect transmitted decisions and make a final decision. Two types of medium access control that the sensor nodes apply to send their decisions are investigated: time division multiple access and slotted-Aloha. Unlike using the time division multiple access protocol, the slotted-Aloha-based distributed detection will experience packet collisions. However, in this article, since only uncensored decisions are sent, packet collisions are informative. We derive fusion rules according to generalized likelihood ratio test, Rao test, and Wald test for both the time division multiple access–based distributed detection and the slotted-Aloha-based distributed detection. We see that the fusion rules for the slotted-Aloha-based distributed detection here also exploit packet collisions in the final decision-making. In addition, the asymptotic performances and energy consumption of both schemes are analyzed. Extensive simulation and numerical results are provided to compare the performances of these two schemes. We show that, for a given time delay, the slotted-Aloha-based distributed detection can outperform the time division multiple access–based distributed detection by increasing the number of sensor nodes which results in higher energy consumption.

Target Detection in Passive Radar Under Noisy Reference Channel: A New Threshold-Setting Strategy

In the detection theory framework, it is customary to assign a bound to the false alarm probability and to attempt to maximize the detection probability subject to this constraint. In the problem of moving target detection in passive radar with a noisy reference channel, we formulate a detection problem as a composite hypothesis-testing problem and solve it with the likelihood ratio test (LRT) principle, which is known as generalized LRT in the electrical engineering works of literature. In such a problem, we show that any uncertainty in the value of the direct signal-to-noise ratio of the reference channel, abbreviated as $\text{DNR}_r$ , can result in excessive false alarm probability of the proposed noisy-reference-channel-based detector in the low- $\text{DNR}_r$ regime. To facilitate efficient operation under uncertainty in $\text{DNR}_r$ , we propose a new threshold-setting strategy to adjust the level of the proposed detector. Through extensive Monte–Carlo simulations, we examine the above problem and investigate the efficiency of the proposed threshold-setting strategy. Besides, we apply the framework of the kernel theory to the target detection problem of a noisy and ideal reference channel passive radar to propose two new detectors. As such, we replace the inner products of the proposed tests with appropriate polynomial kernel functions allowing for richer feature space to be deployed in the detection, achieving better detection performance. In this case, our detection performance results show that the kernelized detectors offer more that 1-dB signal-to-noise ratio gain as compared to their conventional counterparts.

Centralized Versus Decentralized Detection of Attacks in Stochastic Interconnected Systems

We consider a security problem for interconnected systems with linear, discrete, time-invariant, stochastic dynamics, where the objective is to detect exogenous attacks by processing measurements at different locations. We consider centralized and decentralized detectors, which differ primarily in their knowledge of the system model. In particular, a decentralized detector has a model of the dynamics of the isolated subsystems, but is unaware of the interconnection signals that are exchanged among subsystems. Instead, a centralized detector has a model of the entire dynamical system. We characterize the performance of the two detectors and show that, depending on the system and attack parameters, each of the detectors can outperform the other. Hence, it may be possible for the decentralized detector to outperform its centralized counterpart, despite having less information about the system dynamics, and this property is due to the nature of the considered attack detection problem (that is, a simple vs. composite hypothesis testing problem). Finally, we numerically validate our findings on a power system model.

Target Detection With Imperfect Waveform Separation in Distributed MIMO Radar

This paper considers target detection in distributed multiple-input multiple-output (MIMO) radar with imperfect waveform separation at local receivers. The problem is formulated as a binary composite hypothesis testing problem, where target residuals due to imperfect waveform separation are explicitly modeled as a subspace component in the alternative hypothesis, while disturbances including the clutter and thermal noise are present under both hypotheses. Under assumptions of fluctuating and non-fluctuating target amplitude over a scan, e.g., Swerling models, we particularly consider a distributed hybrid-order Gaussian (DHOG) signal model and develop the generalized likelihood ratio test (GLRT) which relies on the maximum likelihood (ML) estimation of the target amplitude and the residual covariance matrix under the alternative hypothesis. The Cramer-Rao bounds (CRBs) on estimating the target amplitude and residual subspace covariance matrix are derived. Simulation results in both local and distributed scenarios confirm the effectiveness of the proposed GLRT and show improved performance in terms of receiver operating characteristic (ROC) by exploiting the existence of target residual component.

Anonymous Heterogeneous Distributed Detection: Optimal Decision Rules, Error Exponents, and the Price of Anonymity

We explore the fundamental limits of heterogeneous distributed detection in an anonymous sensor network with $n$ sensors and a single fusion center. The fusion center collects the single observation from each of the $n$ sensors to detect a binary parameter. The sensors are clustered into multiple groups, and different groups follow different distributions under a given hypothesis. The key challenge for the fusion center is the anonymity of sensors—although it knows the exact number of sensors and the distribution of observations in each group, it does not know which group each sensor belongs to. It is hence natural to consider it as a composite hypothesis testing problem. First, we propose an optimal test called mixture likelihood ratio test , which is a randomized threshold test based on the ratio of the uniform mixture of all the possible distributions under one hypothesis to that under the other hypothesis. Optimality is shown by first arguing that there exists an optimal test that is symmetric , that is, it does not depend on the order of observations across the sensors, and then proving that the mixture likelihood ratio test is optimal among all symmetric tests. Second, we focus on the Neyman–Pearson setting and characterize the error exponent of the worst-case type-II error probability as $n$ tends to infinity, assuming the number of sensors in each group is proportional to $n$ . Finally, we generalize our result to find the collection of all achievable type-I and type-II error exponents, showing that the boundary of the region can be obtained by solving an optimization problem. Our results elucidate the price of anonymity in heterogeneous distributed detection, and can be extended to $M$ -ary hypothesis testing with heterogeneous observations generated according to hidden latent variables.

Spectrum sensing in a calibrated multi-antenna cognitive radio: Exact LRT approaches

This paper addresses the spectrum sensing problem in a calibrated multi-antenna cognitive radio. To do so, a composite hypothesis testing problem is formulated according to the covariance matrix of the received signal. Then, we resort to the likelihood ratio test (LRT) principle to derive some new LRT tests. This problem has been previously solved based on two-step Generalized LRT (2S-GLRT), leading to different detection strategies in comparison to this study; thus the proposed method is referred to as an exact spectrum sensing method. We provide some simulation results to show that the detection performance of the proposed detectors is better than their counterparts, derived based on the 2S-GLRT, as well as some other state-of-the-art approaches given in the literature.

Spectrum Sensing in SIMO Cognitive Radios Under Primary User Transmitter IQ Imbalance

In this paper, we model the received signal of a single-input-multiple-output (SIMO) cognitive radio with in-phase and quadrature imbalance (IQI) in the transmitter. Then, the spectrum sensing problem is formulated as a composite hypothesis testing problem. We propose two new eigenvalue-based detectors based on the likelihood ratio test (LRT), Rao and Wald frameworks, i.e., the resultant LRT leads us to the Wald test. For the proposed Rao test, we find closed-form expressions for false alarm and detection probabilities under noise variance uncertainty in order to get the level of desired false alarm probability $p_{\text{fa}}$ , i.e., it is called a level $p_{\text{fa}}$ test. This is required for nonconstant false alarm rate (non-CFAR) such as the Rao test to guarantee that the obtained false alarm rate is always below a required level $p_{\text{fa}}$ . Additionally, an analytical formula for the probability of false alarm of Wald test is presented to confirm our simulation results. In this case, the proposed Wald detector is a size $p_{\text{fa}}$ test, called a CFAR detector. Detection performances of the proposed detectors are compared in different scenarios by extensive Monte Carlo simulations. It is shown that the Wald detector outperforms the proposed Rao one. Also, we show that the effects of transmitting IQI can be considered as a signal-to-noise ratio factor, just affecting the detection performance. Also, our performance results show that the proposed Wald detector is fairly robust to the receiver IQI when direct down-conversion receivers are exploited.

Cognitive Radio Spectrum Sensing Under Joint TX/RX I/Q Imbalance and Uncalibrated Receiver

In high-speed wireless communication systems, direct-conversion transceivers (DCTRs) are in demand due to low implementation cost of analog front ends. However, the DCTRs are more susceptible to radio frequency impairments like in-phase and quadrature (I/Q) imbalance. This has been shown to significantly limit the spectrum sensing (SS) capability of a secondary user to find spectrum holes. Another source of performance limitation in multiantenna cognitive radio (CR) systems is the uncalibrated receivers across the antennas. Unlike previous works, which considered these two impairments separately, we model the received signal as a single-input multiple-output CR system in the presence of transmit (TX) and receive (RX) I/Q imbalances and when the receiver is uncalibrated across the antennas. In such a scenario, we model the SS problem as a composite binary hypothesis testing problem, apply the likelihood ratio test principle, and develop a new robust SS approach. The performance of the proposed SS algorithm is evaluated using extensive Monte Carlo simulations to illustrate the superiority of the proposed detector in comparison with the existing SS algorithms in the literature. The obtained results indicate that the proposed SS algorithm has constant false alarm rate property with respect to any nonuniform noise variance uncertainties as well as RX I/Q imbalance parameters, thereby being a robust and blind detector.

Adaptive confidence sets for matrix completion

In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.

Thresholds for detecting an anomalous path from noisy environments

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 $\mu_{n}$ on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] showed that if the unknown directed path under the signal hypothesis has known initial location, then detection is possible (in the minimax sense) if $\mu_{n}\gg1/\sqrt{\log n}$, while it is not possible if $\mu_{n}\ll1/\log n\sqrt{\log\log n}$. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757], the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] with a polynomial statistic, which is obtained by employing a multiscale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin [Comm. Math. Phys. 294 (2010) 471–503].

Nonparametric Composite Hypothesis Testing in an Asymptotic Regime

We investigate the nonparametric, composite hypothesis testing problem for arbitrary unknown distributions in the asymptotic regime where both the sample size and the number of hypotheses grow exponentially large. Such asymptotic analysis is important in many practical problems, where the number of variations that can exist within a family of distributions can be countably infinite. We introduce the notion of \emph{discrimination capacity}, which captures the largest exponential growth rate of the number of hypotheses relative to the sample size so that there exists a test with asymptotically vanishing probability of error. Our approach is based on various distributional distance metrics in order to incorporate the generative model of the data. We provide analyses of the error exponent using the maximum mean discrepancy (MMD) and Kolmogorov-Smirnov (KS) distance and characterize the corresponding discrimination rates, i.e., lower bounds on the discrimination capacity, for these tests. Finally, an upper bound on the discrimination capacity based on Fano's inequality is developed. Numerical results are presented to validate the theoretical results.

Robust composite binary hypothesis testing via measure-transformed quasi score test

• This paper handles the problem of composite binary hypothesis testing when an accurate probability model is not available. • Under this framework, a robust generalization of the Gaussian quasi score test is developed. • The proposed generalization, assumes a Gaussian model after applying a transform to the probability measure of the data. • The considered measure-transformation is structured by a non-negative data weighting function, called MT-function. • Proper selection of the MT-function leads to significant mitigation of the model mismatch effect. This paper deals with the problem of composite binary hypothesis testing when an accurate parametric probability model is not available. Under this framework, a robust generalization of the Gaussian quasi score test (GQST) is developed. The proposed generalization, called measure-transformed (MT) GQST assumes a Gaussian probability model after applying a transform to the probability measure (distribution) of the data. The considered measure-transformation is structured by a non-negative data weighting function, called MT-function. By proper selection of the MT-function, we show that, unlike the GQST, the proposed MT-GQST can gain resilience against heavy-tailed noise outliers, leading to significant mitigation of the model mismatch effect (introduced by the normality assumption), and yet, have the implementation advantages of the standard GQST (arising from the convenient Gaussian model). The proposed MT-GQST is applied for testing the vector parameters of linear and nonlinear multivariate data models. Simulation examples illustrate its advantages as compared to the GQST and other robust detectors.