Uniformly Most Powerful Invariant Test Research Articles

Correlation-Based Maximal Invariant Statistic for Spectrum Sensing

In this letter, spectrum sensing for wide-sense stationary signals with temporal correlation is studied. In order to perform uniformly most powerful invariant test (UMPIT) or locally most powerful invariant test (LMPIT), the ratio of the distributions of the maximum invariant statistics is derived in closed-form using Wijsman’s theorem. This ratio shows that UMPIT can be obtained when the signal-to-noise ratio (SNR) and the normalized auto-correlation matrix of the primary user signal are both known. In addition, the LMPIT with nominal SNR (LMPIT-NSNR) and the LMPIT with improved sample auto-correlation matrix (LMPIT-ISAC) are proposed based on this ratio. LMPIT-NSNR and LMPIT-ISAC are suitable for the case where the normalized auto-correlation matrix is known and the SNR is unknown, and for the case where both the normalized auto-correlation matrix and the SNR are unknown, respectively. Simulation results show that the performance of LMPIT-NSNR is robust to the fixed nominal SNR value and can approach the performance of UMPIT, and the LMPIT-ISAC performs better than the traditional covariance-based test.

Invariant subspace detector in distributed multiple‐input multiple output radar: geometry gain helps improving moving target detection

To detect moving targets in a distributed multiple-input multiple output (MIMO) radars, it is preferable to null the clutter rather than whiten the clutter. This is mainly because clutter nulling approaches do not require range training data to form a sample covariance estimate of clutter in each transmit-receive pair, especially in the distributed MIMO radars with non-homogeneous clutter. Moreover, geometry diversity of the distributed MIMO helps improve moving target detection since for a given target velocity, different transmit-receive pairs produce different Doppler frequencies that are less likely to be all small and reside in the clutter nulling region. Based on these facts, the authors consider a subspace model for clutter and derive a uniformly most powerful invariant (UMPI) detector as an optimum invariant test. In this case, analytical expressions for calculating the false alarm and detection probabilities for Swerling 0 and Swerling 1 target models are derived in the closed-forms. Moreover, theoretical and numerical analyses of the proposed UMPI test are represented for several scenarios. More importantly, the simulation results show that the proposed subspace-based UMPI detector can attain the predetermined false alarm probability and superior detection performance in the presence of correlated clutter. Finally, authors consider a situation in which perfect waveform separations at the local receivers are no longer valid, and see the detection performance degradation of the proposed optimal detector designed for ideal waveform separations.

Optimal invariant detection of a monochromatic plane wave with unknown amplitude, frequency, phase, direction of arrival and noise variance

This study deals with the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival in complex white Gaussian noise with unknown variance. Depending on the natural invariance in scale, temporal modulation and spatial modulation of the problem, the uniformly most powerful invariant (UMPI) test is derived by using the statistical invariance principles. However, the UMPI test does not apply unless the signal-to-noise ratio (SNR) is known. However, it provides us with performance bound to evaluate any invariant test's performance when the SNR is unknown. Typically, the generalised likelihood ratio test (GLRT) and locally most powerful invariant (LMPI) test are derived as realisable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Computer simulation examples corroborate the authors analysis and indicate that the GLRT is close to the UMPI bound especially in the low probability-of-false-alarm (P FA) region of the receiver operating characteristic curve while the performance of the LMPI test is close to that of the UMPI test in the low SNR region.

Invariant detection for short-code QPSK DS-SS signals

This paper focuses on the optimal detection of quadrature phase-shift keying (QPSK) direct sequence spread spectrum (DS-SS) signals in additive white Gaussian noise (AWGN) with unknown parameters. We consider the invariant detection problem using the complex Gaussian mixture and the modulo-shift signal model, and derive constant-false-alarm-rate (CFAR) invariant detectors such as uniformly most-powerful invariant (UMPI) test and other sub-optimal invariant tests such as the generalized likelihood ratio test (GLRT). An approximation of the UMPI test under low signal-noise-ratio scenarios yields an asymptotic locally most-powerful invariant (ALMPI) test, which has similar computational complexity as the cycle feature detectors. The ALMPI test is computationally efficient compared with the UMPI and GLRT tests. Moreover, further approximation to the ALMPI test leads to the incoherent weighted multi-cycle detectors, which serve as the performance upper bound of all the spreading sequence period detectors based on second-order cyclostationary statistics. Simulation results demonstrate that the proposed ALMPI test exhibits better detection performance than cycle feature detectors with finite observation samples.

Signal activity detection of phase-shift keying signals

We propose computationally inexpensive and efficient solutions for signal activity detection of phase-shift keying (PSK) signals in additive white Gaussian noise. We consider the complex amplitude of the signal, as well as the information sequence, as the unknown parameters. In addition, the noise variance is assumed unknown. We derive the generalized likelihood ratio test (GLRT) and suggest a computationally efficient implementation thereof. Furthermore, we develop a new inexpensive detector for binary PSK signals, which we will refer to as the generalized energy detector. To evaluate the performance of these detectors, we attempt to derive a uniformly most powerful invariant (UMPI) test as an optimal detector. It turns out that the UMPI test exists only if the signal-to-noise ratio is known. We use this UMPI test in order to obtain an upper-bound performance for the evaluation of invariant detectors, such as the GLRT. Simulation results illustrate and compare the performance and the efficiency of the proposed signal activity detectors

Invariant tests for rapid-fluctuating radar signal detection with unknown arrival time

We study the rapid-fluctuating radar signal detection, where the arrival time of the received signal, i.e., the range cell index of the target, as well as the complex amplitude of the signal and the noise are unknown. We show that even in additive white Gaussian noise (AWGN) environment, uniformly most powerful invariant (UMPI) test does not exist. Instead, the UMPI detector in known SNR is used as an upper performance bound for performance evaluation of any invariant detector performance. In addition, we derive generalized likelihood ratio (GLR) detectors for this signal in AWGN with unknown noise variance and also in clutter with unknown covariance matrix. The GLR test statistic for AWGN represents the ratio of the maximum power over all cells to the total power. The GLRT for a clutter environment is also a power ratio which is the maximum over all range cells of the Euclidean norms of the spatially whitened observed sequence. Simulation results demonstrate that the performance of the proposed GLR test in AWGN is very close to the upper bound performance, i.e., to that of the UMPI test in known SNR.

On the relationship between the GLRT and UMPI tests for the detection of signals with unknown parameters

The generalized likelihood ratio test (GLRT) is widely used in signal processing applications such as image processing, wireless communications, medical imaging, classification, and signal detection. However, the GLRT does not have many known properties, other than that it is invariant, uniformly most powerful invariant (UMPI) for problems that fit the linear model, and asymptotically (N/spl rarr//spl infin/) UMPI in general. Since it is invariant, it belongs to the class of tests for which the UMPI test is optimal. In this paper, we consider a general class of detection problems in which unknown signal parameters imply a problem invariance that can be described analytically by orthogonal subgroups. This invariance is natural for problems with unknown signal parameters and, for example, include those of the matched subspace detectors of Scharf and Friedlander. We derive the GLRT and UMPI detectors for this general signal class for the case of Gaussian noise. An expression is found that relates the two test statistics showing the UMPI statistic to be the sum of two terms, one of which is the GLRT. Using this, we find that the GLRT and UMPI tests are asymptotically equivalent as signal-to-noise ratio (SNR) approaches infinity (or as probability of false alarm approaches zero). These results are illustrated by extending an example given by Nicolls and de Jager to show the analytic relationship between the GLRT and UMPI tests. The results indicate that the performance between the tests becomes close at signal-to-noise ratios (SNRs) associated with operating points of the receiver operating curve that are typically of interest in signal detection applications.

An Invariance property of the generalized likelihood ratio test

The generalized likelihood ratio test (GLRT) is invariant with respect to transformations for which the hypothesis testing problem itself is invariant. This result from the statistics literature is presented in the context of some simple signal models. This is an important property of the GLRT in light of its widespread use and the recent interest in invariant tests applied to signal processing applications. The GLRT is derived for some examples in which the uniformly most powerful invariant (UMPI) test does and does not exist, including one in which the UMPI test exists and is not given by the GLRT.

Statistical validation of simulation models of observable systems

In this paper, for validating computer simulation models of real, observable systems, an uniformly most powerful invariant (UMPI) test is developed from the generalized maximum likelihood ratio (GMLR). This test can be considered as a result of a new approach to solving the Behrens‐Fisher problem when covariance matrices of two multivariate normal populations (compared with respect to their means) are different and unknown. The test is based on invariant statistic whose distribution, under the null hypothesis, does not depend on the unknown (nuisance) parameters. The sample size and threshold of the UMPI test are determined from minimization of the weighted sum of the model builder's risk and the model user's risk. The proposed test could result in the saving of sample items, if the items of the sample are observed sequentially. In this paper, we present the exact form of the proposed curtailed procedure and examine the expected sample size savings under the null hypothesis. The sample size savings can be bounded by a constant, which is independent of the sample size. Tables are given for the expected sample size savings and maximum sample size saving under the null hypothesis for a range of significance levels (α), dimensions (p) and sample sizes (n). The curtailed test considered in this paper represents improvement over the noncurtailed or standard fixed sample tests.

The Effect of Design Imbalance on the Power of the F‐test of a Variance Component in the One‐Way Random Model

AbstractThe ANOVA‐based F‐test used for testing the significance of the random effect variance component is a valid test for an unbalanced one‐way random model. However, it does not have an uniform optimum property. For example, this test is not uniformly most powerful invariant (UMPI). In fact, there is no UMPI test in the unbalanced case (see Khuri, Mathew, and Sinha, 1998).The power of the F‐test depends not only on the design used, but also on the true values of the variance components. As Khuri (1996) noted, we can gain a better insight into the effect of data imbalance on the power of the F‐test using a method for modelling the power in terms of the design parameters and the variance components. In this study, generalized linear modelling (GLM) techniques are used for this purpose. It is shown that GLM, in combination with a method of generating designs with a specified degree of imbalance, is an effective way of studying the behavior of the power of the F‐test in a one‐way random model.

Cost-Risk Analysis in Statistical Validating Simulation Models of Service Processes

In this paper, for validating computer simulation models of service processes, an uniformly most powerful invariant (UMPI) test is developed from the generalized maximum likelihood ratio (GMLR). This test can be considered as a result of a new approach to solving the Behrens-Fisher problem when covariance matrices of two multivariate normal populations (compared with respect to their means) are different and unknown. The test is based on invariant statistic whose distribution, under the null hypothesis, does not depend on the unknown (nuisance) parameters. The sample size and threshold of the UMPI test are determined from minimization of the weighted sum of the model builder's risk and the model user's risk. The proposed test could result in the saving of sample items, if the items of the sample are observed sequentially. In this paper we present the exact form of the proposed curtailed procedure and examine the expected sample size savings under the null hypothesis. The sample size savings can be bounded by a constant, which is independent of the sample size. Tables are given for the expected sample size savings and maximum sample size saving under the null hypothesis for a range of significance levels (α), dimensions (p) and sample sizes (n). The curtailed test considered in this paper represents improvement over the noncurtailed or standard fixed sample tests.

Efficient Tests for an Autoregressive Unit Root

This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.