Composite Hypothesis Testing Research Articles

Improved PHY-layer authentication utilizing multi-modal features for mmWave MIMO UAV-enabled systems

This paper exploits multi-modal Physical (PHY)-layer features in terms of artificial fingerprint, In-phase/Quadrature (IQ) imbalance and Angle of Arrival (AoA) to propose a novel PHY-layer authentication framework for a Millimeter Wave (mmWave) Multiple-Input Multiple-Output (MIMO) Unmanned Aerial Vehicle (UAV)-enabled communication system. First, we resort to the AoA-based spatial fingerprint to effectively address the challenge of channel fingerprint instability induced by high-speed UAV mobility. To further enhance the low discriminability of hardware fingerprints caused by refined manufacturing techniques, artificial Gaussian noise is injected into the transmission signals to assist the receiver in better distinguishing between legitimate and illegitimate UAVs. Then, we jointly combine with inherent IQ imbalance and AoA features to design a hybrid authentication scheme and thus construct a multi-dimensional fingerprint space for a comprehensive characterization of UAV identities. To theoretically evaluate the effectiveness of the proposed authentication framework, the analytical closed-form expressions of performance metrics like false alarm and detection probabilities are also exactly derived based on the statistical signal processing technology and composite hypothesis testing. Finally, we provide large simulation results to validate the correctness and feasibility of the proposed theoretical models, and also discuss the relation between system security and communication service quality under different artificial fingerprint level.

On the use of the cumulant generating function for inference on time series

Innovative inference procedures for analyzing time series data are introduced. The methodology covers density approximation and composite hypothesis testing based on Whittle's estimator, which is a widely applied M-estimator in the frequency domain. Its core feature involves the cumulant generating function of Whittle's score obtained using an approximated distribution of the periodogram ordinates. A testing algorithm not only significantly expands the applicability of the state-of-the-art saddlepoint test, but also maintains the numerical accuracy of the saddlepoint approximation. Connections are made with three other prevalent frequency domain techniques: the bootstrap, empirical likelihood, and exponential tilting. Numerical examples using both simulated and real data illustrate the advantages and accuracy of the saddlepoint methods.

Spectrum sensing in uncalibrated MIMO-based cognitive radios

This paper addresses the problem of spectrum sensing (SS) in cognitive radio networks utilizing multiple-input multiple-output (MIMO) technology, even when the receivers are not calibrated. The study develops a framework by formulating various composite hypothesis testing SS problems based on different assumptions about the parameter space of the alternative hypothesis. To accomplish this, four new detectors are designed using the likelihood ratio test (LRT), Rao, and matrix information geometry (MIG) principles. We demonstrate analytically that three of the proposed detectors maintain a constant false alarm rate (CFAR) despite noise covariance matrix uncertainty. For the non-CFAR detectors, we introduce a novel analytical threshold-setting method to ensure that their false alarm probabilities remain below a predetermined level, thereby making them useful for practical situations. Finally, extensive simulations are conducted to verify the analytical results, assess false alarm regulation, and compare the proposed SS algorithms with existing approaches in terms of detection probability.

Constrained Bayesian method for testing composite hypotheses concerning normal distribution with equal parameters

The problem of testing composite hypotheses with respect to the equal parameters of a normal distribution using the constrained Bayesian method is discussed. Hypotheses are tested using the maximum likelihood and Stein’s methods. The optimality of our decision rule is shown by the following criteria: the mixed directional false discovery rate, the false discovery rate, and the Type I and Type II errors, under the conditions of providing a desired level of constraint. The algorithms for implementing the proposed methods and the computational tools for their application are included. Simulation results show validity of the theoretical results along with their superiority over the classical Bayesian method.

Adaptive bootstrap tests for composite null hypotheses in the mediation pathway analysis.

Mediation analysis aims to assess if, and how, a certain exposure influences an outcome of interest through intermediate variables. This problem has recently gained a surge of attention due to the tremendous need for such analyses in scientific fields. Testing for the mediation effect (ME) is greatly challenged by the fact that the underlying null hypothesis (i.e. the absence of MEs) is composite. Most existing mediation tests are overly conservative and thus underpowered. To overcome this significant methodological hurdle, we develop an adaptive bootstrap testing framework that can accommodate different types of composite null hypotheses in the mediation pathway analysis. Applied to the product of coefficients test and the joint significance test, our adaptive testing procedures provide type I error control under the composite null, resulting in much improved statistical power compared to existing tests. Both theoretical properties and numerical examples of the proposed methodology are discussed.

Non-destructive one-shot device test under step-stress experiment with lognormal lifetime distribution

Dealing with censored data is an important concern in reliability and survival analysis. Non-destructive one-shot devices are an extreme case of interval censoring, wherein we only know if a device has failed or not before an inspection time. Besides, non-destructive one-shot devices are frequently highly reliable with large lifetimes, and then long experimentation times would be needed for inference under normal operating conditions. Alternatively, accelerated life tests (ALTs) shorten the lifetime of the devices by increasing one or more stress factors causing failure. Then, after suitable inference, results can be extrapolated to normal conditions. In particular, step-stress ALT designs increase the stress level at which devices are tested throughout the experiment at some fixed times. Under the non-destructive one-shot device set-up, the number of failures is recorded at some inspection times, including the times of stress change, then resulting in censored data. Among the most popular lifetime distributions used to analyze survival data, the lognormal distribution has hazard function with an increasing–decreasing behavior, which is encountered often in practice as units usually experience early failure and then stabilize over time in terms of performance. However, the classical maximum likelihood estimator (MLE) of parameters of the lognormal lifetime distribution may get highly influenced by data contamination. In this paper we propose a family of divergence-based robust estimators for non-destructive one-shot device step-stress experiments under the lognormal lifetime distribution. Further, from the robust estimators, a generalization of the popular Wald-type test statistic based on the MLE for testing composite null hypothesis is defined, resulting in a robust divergence-based family of test statistics.

Adaptive Jamming Attack Detection Under Noise Uncertainty in mmWave Massive MIMO Systems

This paper addresses jamming attack detection issue in a millimeter Wave (mmWave) massive MIMO system under noise uncertainty. Specifically, we apply the generalized likelihood ratio test (GLRT) to develop a one-step GLRT scheme for detecting jamming attack under the homogeneous and partially homogeneous noise environments, and exploit training data to estimate the unknown noise statistical information and replace it with resulting estimation in deriving GLRT procedure to obtain adaptive GLRT detection scheme. We also design a two-step GLRT scheme, where we first assume the unknown noise statistical information is known, and derive GLRT based on test data, and then replace it by the sample covariance matrix based on training data only to achieve a fully adaptive jamming attack detector. With the help of statistical signal subspace analysis and composite hypothesis testing theories, we further examine the statistical distributions of the jamming attack detection schemes and present the closed-form expressions of false alarm and detection probabilities for the proposed schemes under different noise environments. Finally, we implement extensive simulations to validate the theoretical results and evaluate the detection efficiency under various parameters.

Adaptive Target Detection With Polarimetric FDA-MIMO Radar

The problem of adaptive radar detection with a polarimetric Frequency Diverse Array Multiple-Input Multiple-Output (FDA-MIMO) radar is addressed in this paper. At the design stage, the target detection problem is formulated as a composite hypothesis test, with the unknowns given by the target angle, incremental range (target displacement with respect to the center of the occupied range cell), and scattering matrix, as well as the interference covariance matrix. The formulated detection problem is handled by resorting to sub-optimal design strategies based on the Generalized Likelihood Ratio (GLR) criterion. The resulting detectors demand, under the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\mathrm{{1}}$</tex-math></inline-formula> hypothesis, the solution of a box-constrained optimization problem for which several iterative techniques, i.e., the Linearized Array Manifold (LAM), the Gradient Projection Method (GPM), and the Coordinate Descent (CD) algorithms, are exploited. At the analysis stage, the performance of the proposed architectures, which ensure the bounded CFAR property, is evaluated via Monte Carlo simulations and compared with the benchmarks in both white and colored disturbance.

Restricted Distance-Type Gaussian Estimators Based on Density Power Divergence and Their Applications in Hypothesis Testing

In this paper, we introduce the restricted minimum density power divergence Gaussian estimator (MDPDGE) and study its main asymptotic properties. In addition, we examine it robustness through its influence function analysis. Restricted estimators are required in many practical situations, such as testing composite null hypotheses, and we provide in this case constrained estimators to inherent restrictions of the underlying distribution. Furthermore, we derive robust Rao-type test statistics based on the MDPDGE for testing a simple null hypothesis, and we deduce explicit expressions for some main important distributions. Finally, we empirically evaluate the efficiency and robustness of the method through a simulation study.

How can childhood maltreatment affect post-traumatic stress disorder in adult: Results from a composite null hypothesis perspective of mediation analysis.

A greatly growing body of literature has revealed the mediating role of DNA methylation in the influence path from childhood maltreatment to psychiatric disorders such as post-traumatic stress disorder (PTSD) in adult. However, the statistical method is challenging and powerful mediation analyses regarding this issue are lacking. To study how the maltreatment in childhood alters long-lasting DNA methylation changes which further affect PTSD in adult, we here carried out a gene-based mediation analysis from a perspective of composite null hypothesis in the Grady Trauma Project (352 participants and 16,565 genes) with childhood maltreatment as exposure, multiple DNA methylation sites as mediators, and PTSD or its relevant scores as outcome. We effectively addressed the challenging issue of gene-based mediation analysis by taking its composite null hypothesis testing nature into consideration and fitting a weighted test statistic. We discovered that childhood maltreatment could substantially affected PTSD or PTSD-related scores, and that childhood maltreatment was associated with DNA methylation which further had significant roles in PTSD and these scores. Furthermore, using the proposed mediation method, we identified multiple genes within which DNA methylation sites exhibited mediating roles in the influence path from childhood maltreatment to PTSD-relevant scores in adult, with 13 for Beck Depression Inventory and 6 for modified PTSD Symptom Scale, respectively. Our results have the potential to confer meaningful insights into the biological mechanism for the impact of early adverse experience on adult diseases; and our proposed mediation methods can be applied to other similar analysis settings.

Mediation analysis method review of high throughput data.

High-throughput technologies have made high-dimensional settings increasingly common, providing opportunities for the development of high-dimensional mediation methods. We aimed to provide useful guidance for researchers using high-dimensional mediation analysis and ideas for biostatisticians to develop it by summarizing and discussing recent advances in high-dimensional mediation analysis. The method still faces many challenges when extended single and multiple mediation analyses to high-dimensional settings. The development of high-dimensional mediation methods attempts to address these issues, such as screening true mediators, estimating mediation effects by variable selection, reducing the mediation dimension to resolve correlations between variables, and utilizing composite null hypothesis testing to test them. Although these problems regarding high-dimensional mediation have been solved to some extent, some challenges remain. First, the correlation between mediators are rarely considered when the variables are selected for mediation. Second, downscaling without incorporating prior biological knowledge makes the results difficult to interpret. In addition, a method of sensitivity analysis for the strict sequential ignorability assumption in high-dimensional mediation analysis is still lacking. An analyst needs to consider the applicability of each method when utilizing them, while a biostatistician could consider extensions and improvements in the methodology.

Super-exponential distinguishability of correlated quantum states

In the problem of asymptotic binary i.i.d. state discrimination, the optimal asymptotics of the type I and the type II error probabilities is in general an exponential decrease to zero as a function of the number of samples; the set of achievable exponent pairs is characterized by the quantum Hoeffding bound theorem. A super-exponential decrease for both types of error probabilities is only possible in the trivial case when the two states are orthogonal and hence can be perfectly distinguished using only a single copy of the system. In this paper, we show that a qualitatively different behavior can occur when there is correlation between the samples. Namely, we use gauge-invariant and translation-invariant quasi-free states on the algebra of the canonical anti-commutation relations to exhibit pairs of states on an infinite spin chain with the properties that (a) all finite-size restrictions of the states have invertible density operators and (b) the type I and the type II error probabilities both decrease to zero at least with the speed e^{-nclog n} with some positive constant c, i.e., with a super-exponential speed in the sample size n. Particular examples of such states include the ground states of the XX model corresponding to different transverse magnetic fields. In fact, we prove our result in the setting of binary composite hypothesis testing, and hence, it can be applied to prove super-exponential distinguishability of the hypotheses that the transverse magnetic field is above a certain threshold vs. that it is below a strictly lower value.

SPRT-Based Efficient Best Arm Identification in Stochastic Bandits

This paper investigates the best arm identification (BAI) problem in stochastic multi-armed bandits in the fixed confidence setting. The general class of the exponential family of bandits is considered. The existing algorithms for the exponential family of bandits face computational challenges. To mitigate these challenges, the BAI problem is viewed and analyzed as a sequential composite hypothesis testing task, and a framework is proposed that adopts the likelihood ratio-based tests known to be effective for sequential testing. Based on this test statistic, a BAI algorithm is designed that leverages the canonical sequential probability ratio tests for arm selection and is amenable to tractable analysis for the exponential family of bandits. This algorithm has two key features: (1) its sample complexity is asymptotically optimal, and (2) it is guaranteed to be δ-PAC. Existing efficient approaches focus on the Gaussian setting and require Thompson sampling for the arm deemed the best and the challenger arm. Additionally, this paper analytically quantifies the computational expense of identifying the challenger in an existing approach. Finally, numerical experiments are provided to support the analysis.

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.

A new omnibus test of fit based on a characterization of the uniform distribution

ABSTRACT In this paper, we revisit the classical goodness-of-fit problems for univariate distributions; we propose a new testing procedure based on a characterization of the uniform distribution. Asymptotic theory for the simple hypothesis case is provided in a Hilbert-Space setting, including the asymptotic null distribution as well as values for the first four cumulants of this distribution, which are used to fit a Pearson system of distributions as an approximation to the limit distribution. Numerical results indicate that the null distribution of the test converges quickly to its asymptotic distribution, making the critical values obtained using the Pearson system particularly useful. Consistency of the test is shown against any fixed alternative distribution and we derive the limiting behaviour under fixed alternatives with an application to power approximation. We demonstrate the applicability of the newly proposed test when testing composite hypotheses. A Monte Carlo power study compares the finite sample power performance of the newly proposed test to existing omnibus tests in both the simple and composite hypothesis settings. This power study includes results related to testing for the uniform, normal, Pareto and gamma distributions. The empirical results obtained indicate that the test is competitive. An application of the newly proposed test in financial modelling is also included.

Testing goodness-of-fit and conditional independence with approximate co-sufficient sampling

Goodness-of-fit (GoF) testing is ubiquitous in statistics, with direct ties to model selection, confidence interval construction, conditional independence testing, and multiple testing, just to name a few applications. While testing the GoF of a simple (point) null hypothesis provides an analyst great flexibility in the choice of test statistic while still ensuring validity, most GoF tests for composite null hypotheses are far more constrained, as the test statistic must have a tractable distribution over the entire null model space. A notable exception is co-sufficient sampling (CSS): resampling the data conditional on a sufficient statistic for the null model guarantees valid GoF testing using any test statistic the analyst chooses. But CSS testing requires the null model to have a compact (in an information-theoretic sense) sufficient statistic, which only holds for a very limited class of models; even for a null model as simple as logistic regression, CSS testing is powerless. In this paper, we leverage the concept of approximate sufficiency to generalize CSS testing to essentially any parametric model with an asymptotically efficient estimator; we call our extension “approximate CSS” (aCSS) testing. We quantify the finite-sample Type I error inflation of aCSS testing and show that it is vanishing under standard maximum likelihood asymptotics, for any choice of test statistic. We apply our proposed procedure both theoretically and in simulation to a number of models of interest to demonstrate its finite-sample Type I error and power.

The Influence of Distortions of the Prior Distribution on the Characteristics of the Sequential Test for Composite Hypotheses

The Influence of Distortions of the Prior Distribution on the Characteristics of the Sequential Test for Composite Hypotheses

Detecting associated genes for complex traits shared across East Asian and European populations under the framework of composite null hypothesis testing

BackgroundDetecting trans-ethnic common associated genetic loci can offer important insights into shared genetic components underlying complex diseases/traits across diverse continental populations. However, effective statistical methods for such a goal are currently lacking.MethodsBy leveraging summary statistics available from global-scale genome-wide association studies, we herein proposed a novel genetic overlap detection method called CONTO (COmposite Null hypothesis test for Trans-ethnic genetic Overlap) from the perspective of high-dimensional composite null hypothesis testing. Unlike previous studies which generally analyzed individual genetic variants, CONTO is a gene-centric method which focuses on a set of genetic variants located within a gene simultaneously and assesses their joint significance with the trait of interest. By borrowing the similar principle of joint significance test (JST), CONTO takes the maximum P value of multiple associations as the significance measurement.ResultsCompared to JST which is often overly conservative, CONTO is improved in two aspects, including the construction of three-component mixture null distribution and the adjustment of trans-ethnic genetic correlation. Consequently, CONTO corrects the conservativeness of JST with well-calibrated P values and is much more powerful validated by extensive simulation studies. We applied CONTO to discover common associated genes for 31 complex diseases/traits between the East Asian and European populations, and identified many shared trait-associated genes that had otherwise been missed by JST. We further revealed that population-common genes were generally more evolutionarily conserved than population-specific or null ones.ConclusionOverall, CONTO represents a powerful method for detecting common associated genes across diverse ancestral groups; our results provide important implications on the transferability of GWAS discoveries in one population to others.

Adaptive Radar Detection in the Presence of Missing-data

This article deals with the problem of adaptive radar detection in a missing-data context, where the complete observations (i.e., downstream information loss mechanisms) are characterized by homogeneous Gaussian disturbance with an unknown but possibly structured covariance matrix. The detection problem, formulated as a composite hypothesis test, is tackled by resorting to suboptimal design strategies, leveraging the generalized likelihood ratio criterion demanding appropriate maximum likelihood estimates (MLEs) of the unknowns under both hypotheses. Capitalizing on some possible <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge about the interference covariance matrix structure, the optimization problems involved in the MLE computation are handled by employing the expectation–maximization (EM) algorithm or its expectation–conditional maximization and multicycle EM variants. At the analysis stage, the performance of the devised architectures is assessed both via Monte Carlo simulations and on measured data for some covariance matrix structures of practical interest.