Classical Likelihood Ratio Test Research Articles

A note on asymptotics of classical likelihood ratio tests for high-dimensional normal distributions

For random samples of size n obtained from m-variate normal distributions, we investigate the classical likelihood ratio tests for their means and covariance matrices in the high-dimensional case. Many researchers analyze the test statistics for the case of n going to infinity and m keeping fixed. In the high-dimensional setting, the objective of this paper is to prove that the likelihood ratio test statistics converge in distribution to normal distributions when both m and n go to infinity with m/n→y∈(0,1]. We obtain this conclusion very intuitively by using Lyapunov’s central limit theorem.

A novel approach to detect line emission under high background in high-resolution X-ray spectra

ABSTRACT We develop a novel statistical approach to identify emission features or set upper limits in high-resolution spectra in the presence of high background. The method relies on detecting differences from the background using smooth tests and using classical likelihood ratio tests to characterize known shapes like emission lines. We perform signal detection or place upper limits on line fluxes while accounting for the problem of multiple comparisons. We illustrate the method by applying it to a Chandra LETGS + HRC-S observation of symbiotic star RT Cru, successfully detecting previously known features like the Fe line emission in the 6–7 keV range and the Iridium-edge due to the mirror coating on Chandra. We search for thermal emission lines from Ne x, Fe xvii, O viii, and O vii, but do not detect them, and place upper limits on their intensities consistent with a ≈1 keV plasma. We serendipitously detect a line at 16.93 Å (0.732 keV) that we attribute to photoionization or a reflection component.

Optimal Correlators and Waveforms for Mismatched Detection

We consider the classical Neymann–Pearson hypothesis testing problem of signal detection, where under the null hypothesis ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ H}}_{0}$ </tex-math></inline-formula> ), the received signal is white Gaussian noise, and under the alternative hypothesis ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ H}}_{1}$ </tex-math></inline-formula> ), the received signal includes also an additional non–Gaussian random signal, which in turn can be viewed as a deterministic waveform plus zero–mean, non-Gaussian noise. However, instead of the classical likelihood ratio test detector, which might be difficult to implement, in general, we impose a (mismatched) correlation detector, which is relatively easy to implement, and we characterize the optimal correlator weights in the sense of the best trade-off between the false-alarm error exponent and the missed-detection error exponent. Those optimal correlator weights depend (non-linearly, in general) on the underlying deterministic waveform under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ H}}_{1}$ </tex-math></inline-formula> . We then assume that the deterministic waveform may also be free to be optimized (subject to a power constraint), jointly with the correlator, and show that both the optimal waveform and the optimal correlator weights may take on values in a small finite set of typically no more than two to four levels, depending on the distribution of the non-Gaussian noise component. Finally, we outline an extension of the scope to a wider class of detectors that are based on linear combinations of the correlation and the energy of the received signal.

Gaussian universal likelihood ratio testing

Summary The classical likelihood ratio test based on the asymptotic chi-squared distribution of the log-likelihood is one of the fundamental tools of statistical inference. A recent universal likelihood ratio test approach based on sample splitting provides valid hypothesis tests and confidence sets in any setting for which we can compute the split likelihood ratio statistic, or, more generally, an upper bound on the null maximum likelihood. The universal likelihood ratio test is valid in finite samples and without regularity conditions. This test empowers statisticians to construct tests in settings for which no valid hypothesis test previously existed. For the simple, but fundamental, case of testing the population mean of $d$-dimensional Gaussian data with an identity covariance matrix, the classical likelihood ratio test itself applies. Thus, this setting serves as a perfect test bed to compare the classical likelihood ratio test against the universal likelihood ratio test. This work presents the first in-depth exploration of the size, power and relationships between several universal likelihood ratio test variants. We show that a repeated subsampling approach is the best choice in terms of size and power. For large numbers of subsamples, the repeated subsampling set is approximately spherical. We observe reasonable performance even in a high-dimensional setting, where the expected squared radius of the best universal likelihood ratio test’s confidence set is approximately 3/2 times the squared radius of the classical likelihood ratio test’s spherical confidence set. We illustrate the benefits of the universal likelihood ratio test through testing a nonconvex doughnut-shaped null hypothesis, where a universal inference procedure can have higher power than a standard approach.

Moderate deviation principle for different types of classical likelihood ratio tests

This paper focuses on the likelihood ratio test (LRT) statistics for different hypothesis tests. Assuming that a random sample is from a normal population, we make the sample size n and the dimension p close to infinity and satisfy p < n − c for some 1 ≤ c ≤ 4 . Based on this assumption, the moderate deviation principle (MDP) for the LRT will be given under the null hypothesis. The corresponding numerical simulation results are shown at the end of the paper.

Testing the skewness of skew-normal distribution by Bayes factors

Likelihood ratio test is based on the ratio of the maximum values of likelihood functions on the whole parameter space and the null space, respectively. However, the likelihood ratio test is sometimes intractable such as the skew-normal distribution because of the irregular likelihood behavior. In this paper, we investigate the testing problem concerning the skew parameter based on Bayes factor, which is just the ratio of the integrals of the likelihoods. Like Wilks phenomenon of the classical likelihood ratio, the asymptotic chi-square distributions of some linear functions of the logarithms of Bayes factors are derived. Then selecting appropriate prior, we can construct tests with asymptotically nominal significance level. At the same time, we give the local power of the tests, which is the same as the power of classical likelihood ratio tests. We also provide an exact test based on the Bayes factor with specific prior, which is competitive with existing tests. Simulation studies show that our tests are effective.

A Note on Asymptotics of Classical Likelihood Ratio Tests for High-Dimensional Normal Distributions

A Note on Asymptotics of Classical Likelihood Ratio Tests for High-Dimensional Normal Distributions

A high dimensional nonparametric test for proportional covariance matrices

This work is concerned with testing the proportionality between two high dimensional covariance matrices. Several tests for proportional covariance matrices, based on modifying the classical likelihood ratio test and applicable in high dimension, have been proposed in the literature. Despite their usefulness, they tend to have unsatisfactory performance for nonnormal high dimensional multivariate data in terms of size or power. This article proposes a new high dimensional test by developing a bias correction to the existing test statistic constructed based on a scaled distance measure. The suggested test is nonparametric without requiring any specific parametric distribution such as the normality assumption. It can accommodate scenarios where the data dimension p is greater than the sample size n, namely the “large p, small n” problem. With the aid of tools in modern probability theory, we study theoretical properties of the newly proposed test, which include the asymptotic normality and a power evaluation. We demonstrate empirically that our proposal has good size and power performances for a range of dimensions, sample sizes and distributions in comparison with the existing counterparts.

A new method for multi-sample high-dimensional covariance matrices test based on permutation

For multi-sample covariance testing, the classical likelihood ratio test is often efficient and powerful in low-dimensional normal cases. However, when the dimension is larger than the sample size, it fails to work in practice and theory. This paper proposes a permutation based test to handle high-dimensional covariance testing problem with more than two samples. Numerical studies show that the test controls type I error rate well for both normal and non-normal data. The power performance is also competitive with existing methods. In addition, a real data example of DNA microarray is analyzed for illustration.

EXPONENTIATED DISCRETE WEIBULL DISTRIBUTION FOR CENSORED DATA

This paper further develops the statistical inference procedure of the exponentiated discrete Weibull distribution (EDW) for data with the presence of censoring. This generalization of the discrete Weibull distribution has the advantage of being suitable to model non-monotone failure rates, such as those with bathtub and unimodal distributions. Inferences about EDW distribution are presented using both frequentist and bayesian approaches. In addition, the classical Likelihood Ratio Test and a Full Bayesian Significance Test (FBST) were performed to test the parameters of EDW distribution. The method presented is applied to simulated data and illustrated with a real dataset regarding patients diagnosed with head and neck cancer.

High-dimensional Edgeworth expansion of the determinant of sample correlation matrix and its error bound

The paper considers the asymptotic distribution on the determinant of the high-dimensional sample correlation matrix of the Gaussian population with independent components. In particular, when the dimension p and the sample size N satisfy , and , the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix are obtained by the Edgeworth expansion method. An application of the result to high-dimensional independence test is also proposed, some numerical simulations reveal that the proposed method outperforms the traditional chi-square approximation method and performs as efficient as the method introduced by Jiang and Yang [Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions, Ann. Statist. 41(4) (2013), pp. 2029–2074].

Spherical regression models with general covariates and anisotropic errors

Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example that they are either scalar or are themselves on the unit sphere, and/or that the error distribution is isotropic. In many practical situations, such models are too inflexible. Here, we develop richer parametric spherical regression models in which the covariates can have quite general structure (for example, they may be on the unit sphere, in Euclidean space, categorical or some combination of these) and in which the errors are anisotropic. We consider two anisotropic error distributions—the Kent distribution and the elliptically symmetric angular Gaussian distribution—and two parametrisations of each which enable distinct ways to model how the response depends on the covariates. Various hypotheses of interest, such as the significance of particular covariates, or anisotropy of the errors, are easy to test, for example by classical likelihood ratio tests. We also introduce new model-based residuals for evaluating the fitted models. In the examples we consider, the hypothesis tests indicate strong evidence to favour the novel models over simpler existing ones.

Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

For a p-variate normal distribution with covariance matrix $$ {\varvec{\Sigma }}$$, the standardized generalized variance (SGV) is defined as the positive pth root of $$ |{\varvec{\Sigma }}| $$ and used as a measure of variability. Testing equality of the SGVs, for comparing the variability of multivariate normal distributions with different dimensions, is still regarded as matter of interest. The most classical test for this problem is the likelihood ratio test (LRT). In this article, testing equality of the SGVs of k multivariate normal distributions with possibly unequal dimensions is studied. To test this hypothesis, two approximations for the null distribution of the LRT statistic are proposed based on the well known Welch–Satterthwaite and Bartlett adjustment distribution approximation methods. Furthermore, the high-dimensional behavior of these approximated distributions is also investigated. Through a wide simulation study: first, the performance of the proposed tests with the classical LRT is compared in terms of type I error, power, and alpha adjusted equivalents; second, the robustness of the procedures with respect to departures from normality assumption is evaluated. Finally, the proposed methods are illustrated with two real data examples.

On LR simultaneous test of high-dimensional mean vector and covariance matrix under non-normality

In this paper, we primarily focus on simultaneous testing mean vector and covariance matrix with high-dimensional non-Gaussian data, based on the classical likelihood ratio test. Applying the central limit theorem for linear spectral statistics of sample covariance matrices, we establish new modification for the likelihood ratio test, and find that this modified test converges in distribution to normal distribution, when the dimension p tends to infinity, proportionate to the sample size n under the null hypothesis. Furthermore, we conduct a simulation study to examine the performance of the test and compare it with other tests proposed in past studies. As the simulation results show, our empirical powers are clearly superior to those of other tests in a series of settings.

Multi-Parameter One-Sided Monitoring Tests

ABSTRACTMulti-parameter one-sided hypothesis test problems arise naturally in many applications. We are particularly interested in effective tests for monitoring multiple quality indices in forestry products. Our search reveals that there are many effective statistical methods in the literature for normal data, and that they can easily be used to test hypotheses regarding parameter values permitting asymptotically normal estimators. We find that the classical likelihood ratio test is unsatisfactory, because to control the size, it must cope with the least favorable distributions at the cost of power. In this article, we find a novel way to slightly ease the size control, obtaining a much more powerful test. Simulation confirms that the new test retains good control of the Type I error and is markedly more powerful than the likelihood ratio test as well as many competitors based on normal data. The new method performs well in the context of monitoring multiple quality indices.

Cooperative Spectrum Sensing as Image Segmentation: A New Data Fusion Scheme

Reliable and efficient identification of spectrum access opportunities is crucial for cognitive radio ad hoc networks (CRAHNs). A promising method to improve sensing reliability is CSS, where sensing data from multiple secondary users is combined to facilitate decision making. However, the performance of CSS algorithms in CRAHNs is limited by two challenges. First, the spectrum occupancy status in CRAHNs is spatially heterogeneous. Second, malfunctioning nodes or malicious attackers may falsify the true sensing results. To handle both these challenges, inspired by image segmentation in computer vision, we propose a novel graph-cut CSS algorithm. This algorithm is robust in the presence of false sensing data reports, computationally efficient (with polynomial worst case time complexity), and can be implemented distributively. Theoretically, it can be regarded as a new data fusion scheme that is generalized from classical likelihood ratio tests when sensing information is fully exploited, or from the majority fusion rule, when thresholded sensing information is utilized. Introduction

Statistical inference in constrained latent class models for multinomial data based on $$\phi $$ ϕ -divergence measures

In this paper we explore the possibilities of applying $$\phi $$ -divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum $$\phi $$ -divergence estimators (M $$\phi $$ Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M $$\phi $$ Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M $$\phi $$ Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, $$\phi $$ -divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, $$\phi $$ -divergence measures can handle the problem and we derive a family of $$\phi $$ -divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.

Tests for covariance structures with high-dimensional repeated measurements

In regression analysis with repeated measurements, such as longitudinal data and panel data, structured covariance matrices characterized by a small number of parameters have been widely used and play an important role in parameter estimation and statistical inference. To assess the adequacy of a specified covariance structure, one often adopts the classical likelihood-ratio test when the dimension of the repeated measurements ($p$) is smaller than the sample size ($n$). However, this assessment becomes quite challenging when $p$ is bigger than $n$, since the classical likelihood-ratio test is no longer applicable. This paper proposes an adjusted goodness-of-fit test to examine a broad range of covariance structures under the scenario of “large $p$, small $n$.” Analytical examples are presented to illustrate the effectiveness of the adjustment. In addition, large sample properties of the proposed test are established. Moreover, simulation studies and a real data example are provided to demonstrate the finite sample performance and the practical utility of the test.

Two-stage test of means of unordered pairs.

The problem of testing equality of means of a bivariate normal distribution on the basis of a sample of size n has been considered when the labels of the observations are either missing or not known. The problem may arise in many applied settings, especially in genetics. Classical likelihood ratio test fails here because of identifiability problems. We propose a two-stage testing procedure using a recently developed test in the context of penalized splines. The proposed testing procedure is found to outperform the tests proposed in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

On divergence tests for composite hypotheses under composite likelihood

It is well-known that in some situations it is not easy to compute the likelihood function as the datasets might be large or the model is too complex. In that contexts composite likelihood, derived by multiplying the likelihoods of subjects of the variables, may be useful. The extension of the classical likelihood ratio test statistics to the framework of composite likelihoods is used as a procedure to solve the problem of testing in the context of composite likelihood. In this paper we introduce and study a new family of test statistics for composite likelihood: Composite $$\phi $$ -divergence test statistics for solving the problem of testing a simple null hypothesis or a composite null hypothesis. To do that we introduce and study the asymptotic distribution of the restricted maximum composite likelihood estimate.