Bartlett Correction Research Articles

Correction to: Bartlett corrections for zero-adjusted generalized linear models

Correction to: Bartlett corrections for zero-adjusted generalized linear models

Higher-order asymptotic refinements in a multivariate regression model with general parameterization

This paper derives a general Bartlett correction formula to improve the inference based on the likelihood ratio test in a multivariate model under a quite general parameterization, where the mean vector and the variance-covariance matrix can share the same vector of parameters. This approach includes a number of models as special cases such as non-linear regression models, errors-in-variables models, mixed-effects models with non-linear fixed effects, and mixtures of the previous models. We also employ the Skovgaard adjustment to the likelihood ratio statistic in this class of multivariate models and derive a general expression of the correction factor based on Skovgaard approach. Monte Carlo simulation experiments are carried out to verify the performance of the improved tests, and the numerical results confirm that the modified tests are more reliable than the usual likelihood ratio test. Applications to real data are also presented for illustrative purposes.

Empirical-Likelihood-Based Inference for Partially Linear Models

Partially linear models find extensive application in biometrics, econometrics, social sciences, and various other fields due to their versatility in accommodating both parametric and nonparametric elements. This study aims to establish statistical inference for the parametric component effects within these models, employing a nonparametric empirical likelihood approach. The proposed method involves a projection step to eliminate the nuisance nonparametric component and utilizes an empirical-likelihood-based technique, along with the Bartlett correction, to enhance the coverage probability of the confidence interval for the parameter of interest. This method demonstrates robustness in handling normally and non-normally distributed errors. The proposed empirical likelihood ratio statistic converges to a limiting chi-square distribution under certain regulations. Simulation studies demonstrate that this method provides better inference in terms of coverage probabilities compared to the conventional normal-approximation-based method. The proposed method is illustrated by analyzing the Boston housing data from a real study.

Higher-order asymptotic corrections and their application to the Gamma Variance Model

We present improved methods for calculating confidence intervals and p values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can incorporate uncertainties in parameters that themselves represent uncertainties (informally, “errors on errors”) called the Gamma Variance Model. This model contains fixed parameters, generically denoted by ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, that represent the relative uncertainties in estimates of standard deviations of Gaussian distributed measurements. If the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are small, one can construct confidence intervals and p values using standard asymptotic methods. This is formally similar to the familiar situation of a large data sample, in which estimators for all adjustable parameters have Gaussian distributions. Here we address the important case where the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are not small and as a consequence the first-order asymptotic distributions do not represent a good approximation. We investigate improved test statistics based on the technology of higher-order asymptotics (modified likelihood root and Bartlett correction). The effective application of higher-order corrections removes an important computational barrier to the use of the Gamma Variance Model.

Bartlett corrections for zero-adjusted generalized linear models

Bartlett corrections for zero-adjusted generalized linear models

Some asymptotic inferential aspects of the Kumaraswamy distribution

The Kumaraswamy distribution is doubly limited, continuous, very flexible, and is widely applied in hydrology and related areas. Recently, several families of distributions based on this distribution have emerged. To make a contribution regarding some asymptotic aspects related to the inferential analysis, we derived an analytic expression of order n − 1 / 2 , where n is the sample size, for the skewness coefficient of the distribution of the maximum likelihood estimators of the parameters of the Kumaraswamy distribution. A simulation study and an application are presented to illustrate that, when the sample size is small, the likelihood inferences may not be reliable. We also obtain Bartlett correction factors for the likelihood ratio statistic as well as the results of the bootstrap likelihood ratio test and bootstrap Bartlett correction and present a Monte Carlo simulation study to compare the rejection rates of the tests in question.

Bartlett correction of an independence test in a multivariate Poisson model

We consider a system of dependent Poisson variables, where each variable is the sum of an independent variate and a common variate. It is the common variate that creates the dependence. Within this system, a test of independence may be constructed where the null hypothesis is that the common variate is identically zero. In the present paper, we consider the maximum log likelihood ratio test. For this test, it is well‐known that the asymptotic distribution of the test statistic is an equal mixture of zero and a chi‐square distribution with one degree of freedom. We examine a Bartlett correction of the test, in the hope that we will get better approximation of the nominal size for moderately large sample sizes. A correction of this type is explicitly derived, and its usefulness is explored in a simulation study. For practical purposes, the correction is found to be useful in dimension two, but not in higher dimensions.

The effect of latent and error non-normality on corrections to the test statistic in structural equation modeling

In structural equation modeling, several corrections to the likelihood-ratio model test statistic have been developed to counter the effects of non-normal data. Previous robustness studies investigating the performance of these corrections typically induced non-normality in the indicator variables. However, non-normality in the indicators can originate from non-normal errors or non-normal latent factors. We conducted a Monte Carlo simulation to analyze the effect of non-normality in factors and errors on six different test statistics based on maximum likelihood estimation by evaluating the effect on empirical rejection rates and derived indices (RMSEA and CFI) for different degrees of non-normality and sample sizes. We considered the uncorrected likelihood-ratio model test statistic and the Satorra–Bentler scaled test statistic with Bartlett correction, as well as the mean and variance adjusted test statistic, a scale-shifted approach, a third moment-adjusted test statistic, and an approach drawing inferences from the relevant asymptotic chi-square mixture distribution. The results indicate that the values of the uncorrected test statistic—compared to values under normality—are associated with a severely inflated type I error rate when latent variables are non-normal, but virtually no differences occur when errors are non-normal. Although no general pattern regarding the source of non-normality for all analyzed measures of fit can be derived, the Satorra–Bentler scaled test statistic with Bartlett correction performed satisfactorily across conditions.

Bartlett-corrected tests for varying precision beta regressions with application to environmental biometrics.

Beta regressions are commonly used with responses that assume values in the standard unit interval, such as rates, proportions and concentration indices. Hypothesis testing inferences on the model parameters are typically performed using the likelihood ratio test. It delivers accurate inferences when the sample size is large, but can otherwise lead to unreliable conclusions. It is thus important to develop alternative tests with superior finite sample behavior. We derive the Bartlett correction to the likelihood ratio test under the more general formulation of the beta regression model, i.e. under varying precision. The model contains two submodels, one for the mean response and a separate one for the precision parameter. Our interest lies in performing testing inferences on the parameters that index both submodels. We use three Bartlett-corrected likelihood ratio test statistics that are expected to yield superior performance when the sample size is small. We present Monte Carlo simulation evidence on the finite sample behavior of the Bartlett-corrected tests relative to the standard likelihood ratio test and to two improved tests that are based on an alternative approach. The numerical evidence shows that one of the Bartlett-corrected typically delivers accurate inferences even when the sample is quite small. An empirical application related to behavioral biometrics is presented and discussed.

Small Sample Adjustment for Hypotheses Testing on Cointegrating Vectors

Abstract Johansen’s (2000. “A Bartlett Correction Factor for Tests of on the Cointegrating Relations.” Econometric Theory 16: 740–78) Bartlett correction factor for the LR test of linear restrictions on cointegrated vectors is derived under the i.i.d. Gaussian assumption for the innovation terms. However, the distribution of most data relating to financial variables is fat-tailed and often skewed; there is therefore a need to examine small sample inference procedures that require weaker assumptions for the innovation term. This paper suggests that using the non-parametric bootstrap to approximate a Bartlett-type correction provides a statistic that does not require specification of the innovation distribution and can be used by applied econometricians to perform a small sample inference procedure that is less computationally demanding than it’s analytical counterpart. The procedure involves calculating a number of bootstrap values of the LR test statistic and estimating the expected value of the test statistic by the average value of the bootstrapped LR statistic. Simulation results suggest that the inference procedure has good finite sample property and is less dependent on the parameter space of the data generating process.

Modified Likelihood Ratio Test for Sub-mean Vectors with Two-step Monotone Missing Data in Two-sample Problem

This article deals with the problem of testing for two normal sub-mean vectors when the data set have two-step monotone missing observations. Under the assumptions that the population covariance matrices are equal, we obtain the likelihood ratio test (LRT) statistic. Furthermore, an asymptotic expansion for the null distribution of the LRT statistic is derived under the two-step monotone missing data by the perturbation method. Using the result, we propose two improved statistics with good chi-squared approximation. One is the modified LRT statistic by Bartlett correction,and the other is the modified LRT statistic using the modification coefficient by linear interpolation. The accuracy of the approximations are investigated by using a Monte Carlo simulation. The proposed methods are illustrated using an example.

Test for homogeneity with unordered paired observations

In some applications, an experimental unit is composed of two distinct but related subunits. The response from such a unit is (X1,X2) but we observe only Y1=min{X1,X2} and Y2=max{X1,X2}, i.e., the subunit identities are not observed. We call (Y1,Y2) unordered paired observations. Based on unordered paired observations {(Y1i,Y2i)}i=1n, we are interested in whether the marginal distributions for X1 and X2 are identical. Testing methods are available in the literature under the assumptions that Var(X1)=Var(X2) and COV(X1,X2)=0. However, by extensive simulation studies, we observe that when one or both assumptions are violated, these methods have inflated type I errors or much lower powers. In this paper, we study the likelihood ratio test statistics for various scenarios and explore their limiting distributions without these restrictive assumptions. Furthermore, we develop Bartlett correction formulae for these statistics to enhance their precision when the sample size is not large. Simulation studies and real-data examples are used to illustrate the efficacy of the proposed methods.

Likelihood ratio testing in linear state space models: An application to dynamic stochastic general equilibrium models

This paper considers the problem of hypothesis testing in linear Gaussian state space models. We consider two hypotheses of interest: a simple null and a hypothesis of explicit parameter restrictions. We derive the asymptotic distributions of the corresponding likelihood ratio test statistics and compute the Bartlett adjustments. The results are non-trivial because the unrestricted state space model is not (even locally) identified. We apply our analysis to test the validity of the Dynamic Stochastic General Equilibrium (DSGE) models. A Monte Carlo exercise illustrates our findings and confirms the importance of Bartlett corrections at sample sizes typically encountered in macroeconomics.

POWER PROPERTIES OF EMPIRICAL LIKELIHOOD FOR STATIONARY PROCESSES

In this paper we discuss the second order power properties of empirical likelihood for stationary processes. The asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives is given. 1. Introduction. Since its introduction by Owen (1988, 1990), empirical likelihood has become a useful tool for nonparametric inference. It is well known that the empirical likelihood ratio statistic inherits a number of properties of the parametric likelihood ratio statistic. Owen has shown that the empirical likelihood ratio statistic has limiting chi- squared distribution. Qin and Lawless (1994) connected the theories of empirical likelihood and general estimating equations. Hence using empirical likelihood ratio statistics it is possible to obtain tests and confidence regions for a wide range of problems, including linear models (Owen, 1991). Another property of empirical likelihood which also resembles that of a parametric likelihood is Bartlett correction; see for example Hall and La Scala (1990) for the case of the mean parameters, DiCiccio, et al (1991) for the case of smooth functions of means, Chen (1993, 1994) for linear regression model, Chen and Cui (2006) in the presence of nuisance parameters. For dependent data, Monti (1997) applied the empirical likelihood approach to the derivative of the Whittle likelihood. Kitamura (1997) considered blockwise empirical likeli- hood ratios based on data blocks rather than individual observations. Recently, Nordman and Lahiri (2006) introduced a version of empirical likelihood based on the periodogram and spectral estimating equations. They elucidated the asymptotic properties of frequency domain empirical likelihood for linear processes exhibiting both short- and long-range de- pendence. In this paper we consider the second order power properties of empirical likelihood for stationary processes. Section 2 provides a survey of frequency domain empirical likelihood which is due to Nordman and Lahiri (2006). Section 3 gives the asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives. The proofs of results are relegated to Section 4.

Bootstrap-based testing inference in beta regressions

We address the issue of performing testing inference in small samples in the class of beta regression models. We consider the likelihood ratio test and its standard bootstrap version. We also consider two alternative resampling-based tests. One of them uses the bootstrap test statistic replicates to numerically estimate a Bartlett correction factor that can be applied to the likelihood ratio test statistic. By doing so, we avoid estimation of quantities located in the tail of the likelihood ratio test statistic null distribution. The second alternative resampling-based test uses a fast double bootstrap scheme in which a single second level bootstrapping resample is performed for each first level bootstrap replication. It delivers accurate testing inferences at a computational cost that is considerably smaller than that of a standard double bootstrapping scheme. The Monte Carlo results we provide show that the standard likelihood ratio test tends to be quite liberal in small samples. They also show that the bootstrap tests deliver accurate testing inferences even when the sample size is quite small. An empirical application is also presented and discussed.

Likelihood-based inference in censored exponential regression models

This paper deals with the issue of testing hypotheses in the censored exponential regression model in small and moderate-sized samples. We focus on four tests, namely the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic under a null hypothesis and the corresponding reference chi-squared asymptotic distribution. Bartlett and Bartlett-type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in the class of censored exponential regression models. A Bartlett-type correction for the gradient test is derived in this paper in this class of models. Additionally, we also propose bootstrap-based inferential improvements to the four tests mentioned. We numerically compare the tests through extensive Monte Carlo simulation experiments. The numerical results reveal that the corrected and bootstrapped tests exhibit type I error probability closer to the chosen nominal level with virtually no power loss. We also present an empirical application for illustrative purposes.

Higher-order asymptotic refinements in the multivariate Dirichlet regression model

The likelihood ratio test statistic provides the basis for testing inference on the regression parameters in the class of multivariate Dirichlet regression models, which is very useful in modeling multivariate positive observations summing to one. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of the likelihood ratio statistic. Our simulation results suggest that the likelihood ratio test tends to be extremely liberal when the sample size is small. We derive a general Bartlett correction factor in matrix notation for the likelihood ratio test statistic, which reduces the size distortion of the test, and also consider a bootstrap-based Bartlett correction. We also employ the Skovgaard’s adjustment to the likelihood ratio statistic. We numerically compare the proposed tests with the usual likelihood ratio test. Our simulation results suggest that the proposed corrected tests can be interesting alternatives to usual likelihood ratio test since they lead to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes.

Bartlett correction of frequency domain empirical likelihood for time series with unknown innovation variance

The Bartlett correction is a desirable feature of the likelihood inference, which yields the confidence region for parameters with improved coverage probability. This study examines the Bartlett correction for the frequency domain empirical likelihood (FDEL), based on the Whittle likelihood of linear time series models. Nordman and Lahiri (Ann Stat 34:3019–3050, 2006) showed that the FDEL does not have an ordinary Chi-squared limit when the innovation is non-Gaussian with unknown variance, which restricts the use of the FDEL inference in time series. We show that, by profiling the innovation variance out of the Whittle likelihood function, the FDEL is Chi-squared-distributed and Bartlett correctable. In particular, the order of the coverage error of the confidence region can be reduced from $$O(n^{-1})$$ to $$O(n^{-2})$$ .

On Bartlett correction of empirical likelihood for regularly spaced spatial data

AbstractBartlett correction constitutes one of the attractive features of empirical likelihood because it enables the construction of confidence regions for parameters with improved coverage probabilities. We study the Bartlett correction of spatial frequency domain empirical likelihood (SFDEL) based on general spectral estimating functions for regularly spaced spatial data. This general formulation can be applied to testing and estimation problems in spatial analysis, for example testing covariance isotropy, testing covariance separability as well as estimating the parameters of spatial covariance models. We show that the SFDEL is Bartlett correctable. In particular, the improvement in coverage accuracies of the Bartlett‐corrected confidence regions depends on the underlying spatial structures. The Canadian Journal of Statistics 47: 455–472; 2019 © 2019 Statistical Society of Canada

Improved Small Sample Inference Methods for a Mixed-Effects Model for Repeated Measures Approach in Incomplete Longitudinal Data Analysis

The mixed-effects model for repeated measures (MMRM) approach has been widely applied for longitudinal clinical trials. Many of the standard inference methods of MMRM could possibly lead to the inflation of type I error rates for the tests of treatment effect, when the longitudinal dataset is small and involves missing measurements. We propose two improved inference methods for the MMRM analyses, (1) the Bartlett correction with the adjustment term approximated by bootstrap, and (2) the Monte Carlo test using an estimated null distribution by bootstrap. These methods can be implemented regardless of model complexity and missing patterns via a unified computational framework. Through simulation studies, the proposed methods maintain the type I error rate properly, even for small and incomplete longitudinal clinical trial settings. Applications to a postnatal depression clinical trial are also presented.