Rao's Score Research Articles

On local power of likelihood-based tests in von Mises regressions

The von Mises distribution has played a central role as a distribution on the circle. Its associated circular regression model has been applied in a number of areas. In this paper, we consider the von Mises regression model and, under a sequence of Pitman alternatives, derive the nonnull asymptotic expansions of the cumulative distribution functions of the likelihood ratio, Wald, Rao score, and gradient test statistics for testing a subset of the von Mises regression parameters, as well as for testing the concentration parameter. We then compare analytically the local power of these likelihood-based tests on the basis of the asymptotic expansions and provide conditions where one test can be more locally powerful than the other one in this class of regression models. Consequently, on the basis of the general conditions established, the user can choose the most powerful test to make inferences on the model parameters. We also provide a numerical example to illustrate the usefulness and applicability of the general result.

Testing covariance structures belonging to a quadratic subspace under a doubly multivariate model

AbstractA hypothesis related to the block structure of a covariance matrix under the doubly multivariate normal model is studied. It is assumed that the block structure of the covariance matrix belongs to a quadratic subspace, and under the null hypothesis, each block of the covariance matrix also has a structure belonging to some quadratic subspace. The Rao score and the likelihood ratio test statistics are derived, and the exact distribution of the likelihood ratio test is determined. Simulation studies show the advantage of the Rao score test over the likelihood ratio test in terms of speed of convergence to the limiting chi-square distribution, while both proposed tests are competitive in terms of their power. The results are applied to both simulated and real-life example data.

Discrepancy between structured matrices in the power analysis of a separability test

An important task in the analysis of multivariate data is testing of the covariance matrix structure. In particular, for assessing separability, various tests have been proposed. However, the development of a method of measuring discrepancy between two covariance matrix structures, in relation to the study of the power of the test, remains an open problem. Therefore, a discrepancy measure is proposed such that for two arbitrary alternative hypotheses with the same value of discrepancy, the power of tests remains stable, while for increasing discrepancy the power increases. The basic hypothesis is related to the separable structure of the observation matrix under a doubly multivariate normal model, as assessed by the likelihood ratio and Rao score tests. It is shown that the particular one-parameter method and the Frobenius norm fail in the power analysis of tests, while the entropy and quadratic loss functions can be efficiently used to measure the discrepancy between separable and non-separable covariance structures for a multivariate normal distribution.

Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on λ-th power skewness and kurtosis

Temperature data, like many other measurements in quantitative fields, are usually modelled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution , including tests for normality and Laplacity when λ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two Z-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain Z-scores that are nearly independent and distributed as standard Gaussians, resulting in a distribution valid for any sample size (up to very high precision for ). The case leads to a powerful test of fit for the Laplace() distribution, whose empirical power is superior to all 39 competitors in the literature, over a wide range of 400 alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.

Approximate normality in testing an exchangeable covariance structure under large- and high-dimensional settings

In this paper the Rao score and likelihood ratio tests for hypothesis related to exchangeable structure of multivariate data covariance matrix are studied. Under the assumption of large-dimensionality the normal approximation of the Rao score test statistics distribution is proven as well as the exact and approximate distributions of the likelihood ratio test are derived. Simulation studies show the advantage of the Rao score test over the likelihood ratio test in both studied contexts: type I error and power. Moreover, the Rao score test is available in the case of high-dimensionality, and it is shown that the normal approximation matches well its distribution in this case. Thus, this latter approximation could be recommended for practical use.

Thresholding tests based on affine LASSO to achieve non-asymptotic nominal level and high power under sparse and dense alternatives in high dimension

Thresholding estimators such as the existing square-root and LAD LASSO, and the new affine and GLM LASSO with new link functions, have the ability to set coefficients to zero. They will yield new pivotal statistics which enjoy high power under sparse or dense alternative hypotheses. Under a general formalism, thresholding tests not only recover existing tests such as Rao score test and Fisher nonparametric sign test, but also unveil new tests, for the global/omnibus hypothesis in high dimension in particular. Although pivotal, the new statistics do not have a known distribution, so the critical value of the test is calculated by straightforward Monte Carlo, which yields exact level and high power as illustrated on simulated data.

Doubly multivariate linear models with block exchangeable distributed errors and site-dependent covariates

ABSTRACT The problem of testing the intercept and slope parameters of doubly multivariate linear models with site-dependent covariates using Rao's score test (RST) is studied. The RST statistic is developed for a block exchangeable covariance structure on the error vector under the assumption of multivariate normality. We compare our developed RST statistic with the likelihood ratio test (LRT) statistic. Monte Carlo simulations indicate that the RST statistic is much more accurate than its counterpart LRT statistic and it takes significantly less computation time than the LRT statistic. The proposed method is illustrated with an example of multiple response variables measured on multiple trees in a single plot in an agricultural study.

Power Comparisons in Contingency Tables

It is an important inferential problem to test no association between two binary variables based on data. Tests based on the sample odds ratio are commonly used. We bring in a competing test based on the Pearson correlation coefficient. In particular, the odds ratio does not extend to higher order contingency tables, whereas Pearson correlation does. It is important to understand how Pearson correlation stacks against the odds ratio in 2 x 2 contingency tables. Another measure of association is the canonical correlation. In this paper, we examine how competitive Pearson correlation in relation to odds ratio in terms of power in the binary context, contrasting further with both the Wald Z and Rao score tests. We generated an extensive collection of joint distributions of the binary variables and estimated the power of the tests under each joint alternative distribution based on random samples. The consensus is that none of the tests dominates the other.

Specification Tests for Spatial Panel Data Models

Specification of a model is one of the most fundamental problems in econometrics. In practice, specification tests are generally carried out in a piecemeal fashion, for example, testing the presence of one-effect at a time ignoring the potential presence of other forms of misspecification. Many of the suggested tests in the literature require estimation of complex models and even then those tests cannot account for multiple forms of departures from the model under the null hypothesis. Using Bera and Yoon (1993) general test principle and a spatial panel model framework, we first propose an overall test for all possible misspecification. Then, we derive adjusted Rao's score (RS) tests for random effect, serial correlation, spatial lag and spatial error, which can identify the definite cause(s) of rejection of the basic model and thus aiding in the steps for model revision. For empirical researchers, our suggested procedures provide simple strategies for model specification search employing only the ordinary least squares (OLS) residuals from a standard linear panel regression. Through an extensive simulation study, we evaluate the finite sample performance of our suggested tests and some of the existing procedures. We find that our proposed tests have good finite sample properties both in terms of size and power. Finally, to illustrate the usefulness of our procedures, we provide an empirical application of our test strategy in the context of the convergence theory of incomes of different economies, which is a widely studied empirical problem in macro-economic growth theory. Our empirical illustration reveals the problems in using and interpreting unadjusted tests, and demonstrates how these problems are rectified in using our proposed adjusted tests.

The gradient test and its finite sample size properties in a conditional maximum likelihood and psychometric modeling context

In asymptotic theory, the gradient test proposed by Terrell is a recent likelihood-based hypothesis testing approach which can be considered as an alternative to the well-established trinity of likelihood ratio, Rao score, and Wald tests. The gradient test has not yet entered into the mainstream of applied statistics. This is particularly true for the psychometric context. This research discusses a novel application of the gradient test within the conditional maximum likelihood and the Rasch modeling framework. It also investigates some of its finite sample size properties and compares it with the classical trinity of chi square tests by conducting an extensive Monte Carlo study. The results confirm that the gradient test has its pros and cons.

Local power of some likelihood‐based tests in multivariate Student‐t regression models

The Student‐ distribution has proved to be a useful alternative to the traditional normal distribution, mainly to deal with heavy tails and when robust estimation is desired. We consider the multivariate Student‐ regression model and derive the nonnull asymptotic expansions (under a sequence of Pitman alternatives) of the distribution functions of the likelihood ratio, Wald, Rao score, and gradient test statistics for testing a subset of regression parameters in this class of multivariate regression models. On the basis of the nonnull asymptotic expansions derived, the power of all four tests, which are equivalent to the first order, are compared. We provide conditions in which one test can be more locally powerful than the other one in this class of multivariate regression models and, hence, one can choose the most powerful test based on the general conditions.

Phase II Monitoring of Generalized Linear Profiles Under Different Types of Changes

Various control charts have been proposed to monitor generalized linear profiles in Phase II. However, robustness of the proposed methods in detecting different types and especially different directions of changes is not well-studied in the literature. In real-world applications different kinds of changes such as drift and multiple change are likely to happen which can be isotonic (increasing) or antitonic (decreasing). This paper studies the robustness of Rao Score Test (RST) method, T2, and multivariate exponential weighted moving average (MEWMA) in different types, drift and multiple, and directions of changes. Rao Score Test method also benefits from a change-point detection approach whose performance is studied as well. According to the results, generally RST method shows a better performance in detecting different types of changes. Moreover, the performance of the RST method is robust to direction of the change, while T2 and MEWMA are not ARL-unbiased and show different performances under isotonic and antitonic changes. Therefore, to address this issue, we proposed a bias-reduced estimator to be used in T2. Our results demonstrate that the proposed control chart outperforms T2 and is less biased than T2. Finally, a real-world problem is presented in which aforementioned methods are applied to real data.

DriverML: a machine learning algorithm for identifying driver genes in cancer sequencing studies.

Although rapid progress has been made in computational approaches for prioritizing cancer driver genes, research is far from achieving the ultimate goal of discovering a complete catalog of genes truly associated with cancer. Driver gene lists predicted from these computational tools lack consistency and are prone to false positives. Here, we developed an approach (DriverML) integrating Rao’s score test and supervised machine learning to identify cancer driver genes. The weight parameters in the score statistics quantified the functional impacts of mutations on the protein. To obtain optimized weight parameters, the score statistics of prior driver genes were maximized on pan-cancer training data. We conducted rigorous and unbiased benchmark analysis and comparisons of DriverML with 20 other existing tools in 31 independent datasets from The Cancer Genome Atlas (TCGA). Our comprehensive evaluations demonstrated that DriverML was robust and powerful among various datasets and outperformed the other tools with a better balance of precision and sensitivity. In vitro cell-based assays further proved the validity of the DriverML prediction of novel driver genes. In summary, DriverML uses an innovative, machine learning-based approach to prioritize cancer driver genes and provides dramatic improvements over currently existing methods. Its source code is available at https://github.com/HelloYiHan/DriverML.

Phase II monitoring of the nominal logistic regression profiles based on Wald and Rao score test statistics (a case study in healthcare: diabetic patients)

In many statistical process monitoring (SPM), quality of process or product describes by a relationship between one or more predictor variable(s) and a nominal response variable called nominal logistic regression profile (NLRP). In this paper, two control charts including Wald and Rao score test (RST) is developed to monitor NLRP in Phase II. In addition, to evaluate the parameter estimation effect on the performance of control charts, maximum likelihood estimation (MLE) is developed in Phase I. Performance of the proposed control charts are evaluated using some simulation studies and results show that Wald control chart has better performance than RST chart under different shifts in NLRP parameters. Furthermore, to show the efficiency of the proposed approaches in real application, a case study in health-care is applied. Real case results is also showed that Wald control chart has better performance in faster detecting the out-of-control condition.

Phase II monitoring of the nominal logistic regression profiles based on Wald and Rao score test statistics (a case study in healthcare: diabetic patients)

In many statistical process monitoring (SPM), quality of process or product describes by a relationship between one or more predictor variable(s) and a nominal response variable called nominal logistic regression profile (NLRP). In this paper, two control charts including Wald and Rao score test (RST) is developed to monitor NLRP in Phase II. In addition, to evaluate the parameter estimation effect on the performance of control charts, maximum likelihood estimation (MLE) is developed in Phase I. Performance of the proposed control charts are evaluated using some simulation studies and results show that Wald control chart has better performance than RST chart under different shifts in NLRP parameters. Furthermore, to show the efficiency of the proposed approaches in real application, a case study in health-care is applied. Real case results is also showed that Wald control chart has better performance in faster detecting the out-of-control condition.

Permutation based testing on covariance separability.

Separability is an attractive feature of covariance matrices or matrix variate data, which can improve and simplify many multivariate procedures. Due to its importance, testing separability has attracted much attention in the past. The procedures in the literature are of two types, likelihood ratio test (LRT) and Rao's score test (RST). Both are based on the normality assumption or the large-sample asymptotic properties of the test statistics. In this paper, we develop a new approach that is very different from existing ones. We propose to reformulate the null hypothesis (the separability of a covariance matrix of interest) into many sub-hypotheses (the separability of the sub-matrices of the covariance matrix), which are testable using a permutation based procedure.We then combine the testing results of sub-hypotheses using the Bonferroni and two-stage additive procedures. Our permutation based procedures are inherently distribution free; thus it is robust to non-normality of the data. In addition, unlike the LRT, they are applicable to situations when the sample size is smaller than the number of unknown parameters in the covariance matrix. Our numerical study and data examples show the advantages of our procedures over the existing LRT and RST.

Robust LM tests for spatial dynamic panel data models

In this study, we introduce adjusted Rao's score test statistics (Lagrange multiplier (LM) tests) for a spatial dynamic panel data (SDPD) model that includes a contemporaneous spatial lag, a time lag and a spatial-time lag. The maximum likelihood estimator for the estimation of SDPD models can have asymptotic bias because of individual and time fixed effects. Bias arises since the limiting distributions of the score functions derived from the corresponding concentrated log-likelihood functions are not centered on zero. First, we show how the score functions should be adjusted to avoid the effect of asymptotic bias on the standard LM test statistics. Second, we further adjust score functions such that the resulting LM test statistics are valid when there is local parametric misspecification in the alternative model. Our adjusted LM test statistics can be used to test the presence of the contemporaneous spatial lag, time lag and spatial-time lag in an SDPD model. In a Monte Carlo study, we demonstrate that our suggested test statistics have good finite sample size and power properties. Finally, we illustrate implementation of these tests in an application on public capital productivity in 48 contiguous US states.

Testing spatial regression models under nonregular conditions

In time series context, estimation and testing issues with autoregressive and moving average (ARMA) models are well understood. Similar issues in the context of spatial ARMA models for the disturbance of the regression, however, remain largely unexplored. In this paper, we discuss the problems of testing no spatial dependence in the disturbances against the alternative of spatial ARMA process incorporating the possible presence of spatial dependence in the dependent variable. The problems of conducting such a test are twofold. First, under the null hypothesis, the nuisance parameter is not identified, resulting in a singular information matrix (IM), which is a nonregular case in statistical inference. To take account of singular IM, we follow Davies (Biometrika 64(2):247–254, 1977; Biometrika 74(1):33–43, 1987) and propose a test procedure based on the supremum of the Rao score test statistic. Second, the possible presence of spatial lag dependence will have adverse effect on the performance of the test. Using the general test procedure of Bera and Yoon (Econom Theory 9:649–658, 1993) under local misspecification, we avoid the explicit estimation of the spatial autoregressive parameter. Thus our suggested tests are entirely based on ordinary least squares estimation. Tests suggested here can be viewed as a generalization of Anselin et al. (Reg Sci Urban Econ 26:77–104, 1996). We conduct Monte Carlo simulations to investigate the finite sample properties of the proposed tests. Simulation results show that our tests have good finite sample properties both in terms of size and power, compared to other tests in the literature. We also illustrate the applications of our tests through several data sets.

On Inference about Tilt Parameter in Marshall-Olkin Family of Distributions

Marshall and Olkin [Biometrika 84, 641–652, 1997] introduced a method for constructing a new distribution by adding a new parameter, called tilt parameter, to a parent distribution. It is observed that adding this parameter leads to a more flexible model than the parent model. In this paper, different estimators for tilt parameter as a major parameter are presented. Their performances are compared using Monte Carlo simulations. Hypothesis testing and interval estimation of tilt parameter using Rao score test is discussed.

Simple tests for endogeneity of spatial weights matrices

In this study, we propose a Rao's score (RS) statistic (Lagrange multiplier (LM) statistic) to test for endogeneity of the spatial weights matrix in a spatial autoregressive model. To achieve this, we start with a spatial autoregressive model with an acceptable form for the generating process for the elements of the endogenous spatial weights matrix as in Qu and Lee (2015). Our test statistic is simple to calculate because it requires computationally simple estimations. By construction, the test statistic is robust in the sense that its asymptotic null distribution is a centered chi-square distribution regardless of the (local) presence of a spatial autoregressive parameter in the alternative model. We summarize the asymptotic properties of our test statistic under the null and the alternative hypotheses. To investigate its finite sample size and power properties, we conduct a Monte Carlo study. The results are in line with our theoretical findings and indicate that the robust test has good size and power properties.