Corrected Likelihood Ratio Test Research Articles

Nonlinear shrinkage test on a large‐dimensional covariance matrix

This paper is concerned with deriving a new test on a covariance matrix which is based on its nonlinear shrinkage estimator. The distribution of the test statistic is deduced under the null hypothesis in the large‐dimensional setting, that is, when with variables and samples both tending to infinity. The theoretical results are illustrated by means of an extensive simulation study where the new nonlinear shrinkage‐based test is compared with existing approaches, in particular with the commonly used corrected likelihood ratio test, the corrected John test, and the test based on the linear shrinkage approach. It is demonstrated that the new nonlinear shrinkage test possesses better power properties under heteroscedastic alternative.

Testing for independence of large dimensional vectors

In this paper new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose we study the weak convergence of linear spectral statistics of central and (conditionally) non-central Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) non-central Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

Improved heteroskedasticity likelihood ratio tests in symmetric nonlinear regression models

In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609–620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423–437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

Optimal hypothesis testing for high dimensional covariance matrices

This paper considers testing a covariance matrix $\Sigma$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_0:\Sigma=\Sigma_0$ for a given covariance matrix $\Sigma_0$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the $U$-statistic based test is also analyzed when $p/n$ is unbounded.

Testing the independence of sets of large-dimensional variables

This paper proposes the corrected likelihood ratio test (LRT) and large-dimensional trace criterion to test the independence of two large sets of multivariate variables of dimensions p1 and p2 when the dimensions p = p1 + p2 and the sample size n tend to infinity simultaneously and proportionally. Both theoretical and simulation results demonstrate that the traditional χ2 approximation of the LRT performs poorly when the dimension p is large relative to the sample size n, while the corrected LRT and large-dimensional trace criterion behave well when the dimension is either small or large relative to the sample size. Moreover, the trace criterion can be used in the case of p > n, while the corrected LRT is unfeasible due to the loss of definition.

A linear model-based test for the heterogeneity of conditional correlations

Current methods of testing the equality of conditional correlations of bivariate data on a third variable of interest (covariate) are limited due to discretizing of the covariate when it is continuous. In this study, we propose a linear model approach for estimation and hypothesis testing of the Pearson correlation coefficient, where the correlation itself can be modeled as a function of continuous covariates. The restricted maximum likelihood method is applied for parameter estimation, and the corrected likelihood ratio test is performed for hypothesis testing. This approach allows for flexible and robust inference and prediction of the conditional correlations based on the linear model. Simulation studies show that the proposed method is statistically more powerful and more flexible in accommodating complex covariate patterns than the existing methods. In addition, we illustrate the approach by analyzing the correlation between the physical component summary and the mental component summary of the MOS SF-36 form across a fair number of covariates in the national survey data.

Fatores clínico-patológicos de predição do acometimento axilar em pacientes com metástases de câncer de mama no linfonodo sentinela

To evaluate which clinical, pathological or immunohistochemical factors may be predictive of metastatic involvement of other lymph nodes in patients with breast carcinoma undergoing sentinel lymph node biopsy (SLNB). A retrospective study carried out with 1,000 successive patients with SLNB from 1998 to 2008. Age, tumor size, histological grade, lymphovascular invasion, hormone receptor status and HER-2, size of metastasis and number of positive SLN were evaluated. The associations between the characteristics of the tumors and the types of metastases were evaluated through chi(2) corrected likelihood ratio tests for insufficient samples. Mean age was 57.6 years and mean tumor size was 1.85 cm. A total of 72.2% SLN were negative and 27.8% were positive, but in 61.9% of the cases, the SLN was the only positive one, with 78.4% having macrometastases, 17.3% micrometastases and 4.3% isolated tumor cells (CTI). Tumor size was predictive of metastases in non-sentinel lymph nodes. After 54 months of follow-up, there were no recurrences in patients with CTI, but one local recurrence and two systemic recurrences were observed in the micrometastasis group, as well as four local and 30 distant metastases in the macrometastasis group. Among the clinical parameters studied, only tumor size was correlated with metastatic involvement in axillary lymph nodes. The size of the metastases and the number of positive SLN also directly increased the possibility of systemic recurrence. The different rates of recurrence indicate that the biological significance of these types of metastases is different and that patients with SLN metastases may also have different risks of metastatic involvement of other axillary lymph nodes.

Bartlett adjustments for two-parameter exponential family models

In this paper we derive a general closed-form expression for the Bartlett correction for testing a scalar parameter of a two-parameter exponential family model. The correction has the advantage for algebraical and numerical purposes since it involves only trivial operations on suitably defined functions. The formula derived is general enough to cover many important and commonly used distributions. Simulations show that the corrected likelihood ratio tests have empirical sizes closer to the nominal size than the classical uncorrected tests.

Approximate Conditional Unit Root Inference

Based on Cox and Reid (1987) adjustments of likelihood ratio (LR) tests for unit roots in higher‐order autoregressive models are proposed. While unit root inference does not fit directly into the framework of Cox and Reid, the ideas are applied in models with multi‐dimensional parameters of interest and only asymptotic orthogonality of parameters. The adjustments are very simple to apply in that they are of the degrees of freedom type. Detailed Monte Carlo experiments reveal that, for a wide range of admissible parameter values, adjusted LR statistics approximate the asymptotic percentiles of the unit root distributions at a much faster rate than unadjusted ones. In addition, the proposed adjustments are compared with simulated Bartlett type corrected LR tests. They behave equally well in a reasonable parameter region, while both fail on the boundary of the parameter region where an additional unit root is introduced.

Corrected likelihood ratio tests for von mises regression models

We present, in matrix notation, a general Bartlett correction formula to improve likelihood ratio tests in von Mises regression models with concentration covariates. The formula is simple enough to be used analytically to obtain several closed-form Bartlett corrections in a variety of important tests. It also has advantages for numerical purposes. Finally, we apply our main result to a few special models.

The Local Power of the Tests of Overidentifying Restrictions

The local power function of the size-corrected likelihood ratio, linearized likelihood ratio, and Lagrange multiplier tests of overidentifying restrictions on a structural equation is the same to the order 1/T. Moreover, this local power function doe s not depend on the k-class estimator used in the calculation of the test statistic. When the author does not use size-corrected tests, a degrees of freedom corrected likelihood ratio test seems to have the best size and power properties. Finally, the implicit null hypothesis of these tests indicates that they can be interpreted as testing the validity of the structural specification of the equation against any other identified structural equation that encompasses the original e quation. Copyright 1988 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

A Monte Carlo investigation of the accuracy of multivariate CAPM tests

Abstract In a multivariate regression model relating individual returns to the market return, CAPM implies non-linear restrictions on the parameters. Several asymptotically valid tests of these restrictions have been suggested. The existing Monte Carlo evidence shows that some of these tests are unreliable for reasonable sample sizes, but does not indicate well which tests are reliable. This paper reports the results of an extensive Monte Carlo experiment. Shanken's CSR test and Jobson and Korkie's corrected likelihood ratio test are quite accurate in all cases we consider.