Approximate Inference for the Factor Loading of a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. Two approximate conditional inference procedures for the factor loading are developed. The first proposal is a very simple procedure but it is not very accurate. The second proposal gives extremely accurate results even for very small sample size. Moreover, the calculations require only the signed log-likelihood ratio statistic and a measure of the standardized maximum likelihood departure. Simulations are used to study the accuracy of the proposed procedures.     

A note on empirical likelihoods derived from pairwise score functions

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution and which are useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large data sets with relative small sample sizes. Two Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Linear regression analysis with inequality constraints on the regression parameters via empirical likelihood

An empirical likelihood ratio test is developed for testing for or against inequality constraints on regression parameters in linear regression analysis. The proposed approach imposes no parametric model nor identically distributing assumption on the random errors. The asymptotic distribution of the proposed test statistic under null hypothesis is shown to be of chi-bar-squared type. The asymptotic power under contiguous alternatives is also briefly discussed. Moreover, an adjusted empirical likelihood method is adopted to improve the small sample size behaviour of the proposed test. Several simulation studies are carried out to assess the finite sample performance of the proposed tests. The results reveal that the proposed tests could be valuable for improving inference efficiency. A real-life example is discussed to illustrate the theoretical results.     

Profile Likelihood Inferences on the Partially Linear Model with a Diverging Number of Parameters

In this article, we study the profile likelihood estimation and inference on the partially linear model with a diverging number of parameters. Polynomial splines are applied to estimate the nonparametric component and we focus on constructing profile likelihood ratio statistic to examine the testing problem for the parametric component in the partially linear model. Under some regularity conditions, the asymptotic distribution of profile likelihood ratio statistic is proposed when the number of parameters grows with the sample size. Numerical studies confirm our theory.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

Modified likelihood ratio statistics for inflated beta regressions

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are fairly simple since they only require first- and second-order log-likelihood cumulants. Simulation results show that the modified likelihood ratio tests deliver much accurate inference in small samples. An empirical application is presented and discussed.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Likelihood Based Inference in Non-linear Regression Models Using the p* and R* Approach

We develop second order asymptotic results for likelihood-based inference in Gaussian non-linear regression models. We provide an approximation to the conditional density of the maximum likelihood estimator given an approximate ancillary statistic (the affine ancillary). From this approximation, we derive a statistic to test an hypothesis on one component of the parameter. This test statistic is an adjustment of the signed log-likelihood ratio statistic. The distributional approximations (for the maximum likelihood estimator and for the test statistic) are of second order in large deviation regions.     

Improved likelihood inference in beta regression

We consider the issue of performing accurate small-sample likelihood-based inference in beta regression models, which are useful for modelling continuous proportions that are affected by independent variables. We derive small-sample adjustments to the likelihood ratio statistic in this class of models. The adjusted statistics can be easily implemented from standard statistical software. We present Monte Carlo simulations showing that inference based on the adjusted statistics we propose is much more reliable than that based on the usual likelihood ratio statistic. A real data example is presented.     

Empirical Likelihood Method for General Additive-Multiplicative Hazard Models

An empirical likelihood-based inferential procedure is developed for a class of general additive-multiplicative hazard models. The proposed log-empirical likelihood ratio test statistic for the parameter vector is shown to have a chi-squared limiting distribution. The result can be used to make inference about the entire parameter vector as well as any linear combination of it. The asymptotic power of the proposed test statistic under contiguous alternatives is discussed. The method is illustrated by extensive simulation studies and a real example.     

A corrected likelihood ratio statistic for the multivariate regression model with antedependent errors

This paper addresses the problem of inference for the antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as an autoregressive process of general order which is non-stationary in time. Its primary application is in the analysis of repeated measurements data, that is, data consisting of independent replicates of relatively short time series. Our focus is on testing a general linear hypothesis in the context of a multivariate regression model with multivariate normal antedependent errors. Although the relevant likelihood ratio statistic was first presented by Gabriel (1961), the distribution of this statistics has not yet been derived. We present this derivation and show how this result leads to a simple correction factor to improve the x² approximation of the likelihood ratio statistic.     

A modified signed likelihood ratio test in elliptical structural models

In this paper we deal with the issue of performing accurate testing inference on a scalar parameter of interest in structural errors-in-variables models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as special case. We derive a modified signed likelihood ratio statistic that follows a standard normal distribution with a high degree of accuracy. Our Monte Carlo results show that the modified test is much less size distorted than its unmodified counterpart. An application is presented.     

Statistical Inference in Partially Linear Varying-Coefficient Models with Missing Responses at Random

This article considers statistical inference for partially linear varying-coefficient models when the responses are missing at random. We propose a profile least-squares estimator for the parametric component with complete-case data and show that the resulting estimator is asymptotically normal. To avoid to estimate the asymptotic covariance in establishing confidence region of the parametric component with the normal-approximation method, we define an empirical likelihood based statistic and show that its limiting distribution is chi-squared distribution. Then, the confidence regions of the parametric component with asymptotically correct coverage probabilities can be constructed by the result. To check the validity of the linear constraints on the parametric component, we construct a modified generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Then, we extend the generalized likelihood ratio technique to the context of missing data. Finally, some simulations are conducted to illustrate the proposed methods.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Converting observed likelihood to levels of significance for transformation models

Inference for a scalar parameter in the pressence of nuisance parameters requires high dimensional integrations of the joint density of the pivotal quantities. Recent development in asymptotic methods provides accurate approximations for significance levels and thus confidence intervals for a scalar component parameter. In this paper, a simple, efficient and accurate numerical procedure is first developed for the location model and is then extended to the location-scale model and the linear regression model. This numerical procedure only requires a fine tabulation of the parameter and the observed log likelihood function, which can be either the full, marginal or conditional observed log likelihood function, as input and output is the corresponding significance function. Numerical results showed that this approximation is not only simple but also very accurate. It outperformed the usual approximations such as the signed likelihood ratio statistic, the maximum likelihood estimate and the score statistic.     

On the distribution of the likelihood ratio test of equality of normal populations

It has recently been shown by Perlman (1980) that when testing the equality of several normal distributions it is the likelihood ratio test which is unbiased rather than a test based on a modified statistic in common use. This paper gives expansions for the null distribution of the likelihood ratio statistic as well as for the nonnull distribution in a special case.     

Empirical and simulated adjustments of composite likelihood ratio statistics

Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi-square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications, sensitivity and variability matrices usually need to be estimated, but there are no comparisons of the performance of composite likelihood-based statistics in such an instance. A comparison of the accuracy of inference based on the statistics considering two methods typically employed for estimation of sensitivity and variability matrices, namely an empirical method that exploits independent observations, and Monte Carlo simulation, is performed. The results in two examples involving the pairwise likelihood show that a very large number of independent observations should be available in order to obtain accurate coverages using empirical estimation, while limited simulation from the full model provides accurate results regardless of the availability of independent observations. This suggests the latter as a default choice, whenever simulation from the model is possible.