Inference after separated hypotheses testing: an empirical investigation for linear models

Model selection problems arise while constructing unbiased or asymptotically unbiased estimators of measures known as discrepancies to find the best model. Most of the usual criteria are based on goodness-of-fit and parsimony. They aim to maximize a transformed version of likelihood. For linear regression models with normally distributed error, the situation is less clear when two models are equivalent: are they close to or far from the unknown true model? In this work, based on stochastic simulation and parametric simulation, we study the results of Vuong's test, Cox's test, Akaike's information criterion, Bayesian information criterion, Kullback information criterion and bias corrected Kullback information criterion and the ability of these tests to discriminate between non-nested linear models.     

Empiricial Comparison between Some Model Selection Criteria

Model selection aims to find the best model. Most of the usual criteria are based on goodness of fit and parsimony and aim to maximize a transformed version of likelihood. The situation is less clear when two models are equivalent: are they close to the unknown true model or are they far from it? Based on simulations, we study the results of Vuong's test, Cox's test, AIC and BIC and the ability of these four tests to discriminate between models.     

Generalized autoregressive and moving average models: multicollinearity,interpretation and a new modified model

In this paper, we call attention of two observed features in practical applications of the Generalized Autoregressive Moving Average (GARMA) model due to the structure of its linear predictor. One is the multicollinearity which may lead to a non-convergence of the maximum likelihood, using iteratively reweighted least squares, and the inflation of the estimator's variance. The second is that the inclusion of the same lagged observations into the autoregressive and moving average components confounds the interpretation of the parameters. A modified model, GAR-M, is presented to reduce the multicollinearity and to improve the interpretation of the parameters. The expectation and variance under stationarity conditions are presented for the identity and logarithm link function. In a general sense, simulation studies show that the maximum likelihood estimators based on the GARMA and GAR-M models are equivalent but the GAR-M estimators presented a little lower standard errors and some restrictions in the parametric space are imposed to guarantee the stationarity of the process. Also, a real data analysis illustrates the GAR-M fit for daily hospitalization rates of elderly people due to respiratory diseases from October 2012 to April 2015 in São Paulo city, Brazil.     

Modified maximum likelihood estimator under the Jones and Faddy's skew t-error distribution for censored regression model

It is well-known that classical Tobit estimator of the parameters of the censored regression (CR) model is inefficient in case of non-normal error terms. In this paper, we propose to use the modified maximum likelihood (MML) estimator under the Jones and Faddy''s skew t-error distribution, which covers a wide range of skew and symmetric distributions, for the CR model. The MML estimators, providing an alternative to the Tobit estimator, are explicitly expressed and they are asymptotically equivalent to the maximum likelihood estimator. A simulation study is conducted to compare the efficiencies of the MML estimators with the classical estimators such as the ordinary least squares, Tobit, censored least absolute deviations and symmetrically trimmed least squares estimators. The results of the simulation study show that the MML estimators work well among the others with respect to the root mean square error criterion for the CR model. A real life example is also provided to show the suitability of the MML methodology.     

On the efficiency of regression analysis with AR(p) errors

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.     

Likelihood-Based Inference for Weak Exogeneity in I(2) Cointegrated VAR Models

This article develops limit theory for likelihood analysis of weak exogeneity in I(2) cointegrated vector autoregressive (VAR) models incorporating deterministic terms. Conditions for weak exogeneity in I(2) VAR models are reviewed, and the asymptotic properties of conditional maximum likelihood estimators and a likelihood-based weak exogeneity test are then investigated. It is demonstrated that weak exogeneity in I(2) VAR models allows us to conduct asymptotic conditional inference based on mixed Gaussian distributions. It is then proved that a log-likelihood ratio test statistic for weak exogeneity in I(2) VAR models is asymptotically χ² distributed. The article also presents an empirical illustration of the proposed test for weak exogeneity using Japan's macroeconomic data.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Likelihood-based estimation for Gaussian MRFs

Markov random fields (MRFs) express spatial dependence through conditional distributions, although their stochastic behavior is defined by their joint distribution. These joint distributions are typically difficult to obtain in closed form, the problem being a normalizing constant that is a function of unknown parameters. The Gaussian MRF (or conditional autoregressive model) is one case where the normalizing constant is available in closed form; however, when sample sizes are moderate to large (thousands to tens of thousands), and beyond, its computation can be problematic. Because the conditional autoregressive (CAR) model is often used for spatial-data modeling, we develop likelihood-inference methodology for this model in situations where the sample size is too large for its normalizing constant to be computed directly. In particular, we use simulation methodology to obtain maximum likelihood estimators of mean, variance, and spatial-depencence parameters (including their asymptotic variances and covariances) of CAR models.     

Testing non-nested log-linear models with pseudo estimator

As the number of random variables for the categorical data increases, the possible number of log-linear models which can be fitted to the data increases rapidly, so that various model selection methods are developed. However, we often found that some models chosen by different selection criteria do not coincide. In this paper, we propose a comparison method to test the final models which are non-nested. The statistic of Cox (1961, 1962) is applied to log-linear models for testing non-nested models, and the Kullback-Leibler measure of closeness (Pesaran 1987) is explored. In log-linear models, pseudo estimators for the expectation and the variance of Cox's statistic are not only derived but also shown to be consistent estimators.     

A new geometric first-order integer-valued autoregressive (NGINAR(1)) process

A new stationary first-order integer-valued autoregressive process with geometric marginal distributions is introduced based on negative binomial thinning. Some properties of the process are established. Estimators of the parameters of the process are obtained using the methods of conditional least squares, Yule–Walker and maximum likelihood. Also, the asymptotic properties of the estimators are derived involving their distributions. Some numerical results of the estimators are presented with a discussion to the obtained results. Real data are used and a possible application is discussed.     

Hypothesis Testing for Various Familial Dependence Structures

This work compares various hypothesis testing procedures in the case of familial clustered data. Specifically, we use likelihood ratio and Wald's tests for maximum likelihood estimators, and Wald-type tests for moment and quasi-least squares estimators. Using simulations, we estimate significance levels for various hypotheses concerning the one-parent auto-regressive and two-parent equi-correlated dependence structures. We show that the likelihood ratio test performs best for certain simple hypotheses in the one-parent case, whereas the Wald-type test for the quasi-least squares procedure is optimal in the more complex two-parent case.     

Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Inference for the Extended Bifurcating Autoregressive Model for Cell Lineage Studies

Huggins & Basawa (1999) proposed several extensions of the bifurcating autoregressive model used to model cell lineage trees. These models overcame limitations in the original bifurcating autoregressive mode by allowing larger correlations between cousin cells and other cells in the same generation. Huggins & Basawa only considered maximum likelihood inference based on independent trees. This paper examines the asymptotic properties of maximum likelihood estimators based on a single large tree.     

Kendall's tau for serial dependence

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first‐order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Hypothesis testing in mixture regression models

Summary.  We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of n ^−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p -value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Comparison of Selected Methods for Modeling of Multi-State Disease Progression Processes: A Simulation Study

Prognostic studies are essential to understand the role of particular prognostic factors and, thus, improve prognosis. In most studies, disease progression trajectories of individual patients may end up with one of mutually exclusive endpoints or can involve a sequence of different events.

One challenge in such studies concerns separating the effects of putative prognostic factors on these different endpoints and testing the differences between these effects.

In this article, we systematically evaluate and compare, through simulations, the performance of three alternative multivariable regression approaches in analyzing competing risks and multiple-event longitudinal data. The three approaches are: (1) fitting separate event-specific Cox's proportional hazards models; (2) the extension of Cox's model to competing risks proposed by Lunn and McNeil; and (3) Markov multi-state model.

The simulation design is based on a prognostic study of cancer progression, and several simulated scenarios help investigate different methodological issues relevant to the modeling of multiple-event processes of disease progression. The results highlight some practically important issues. Specifically, the decreased precision of the observed timing of intermediary (non fatal) events has a strong negative impact on the accuracy of regression coefficients estimated with either the Cox's or Lunn-McNeil models, while the Markov model appears to be quite robust, under the same circumstances. Furthermore, the tests based on both Markov and Lunn-McNeil models had similar power for detecting a difference between the effects of the same covariate on the hazards of two mutually exclusive events. The Markov approach yields also accurate Type I error rate and good empirical power for testing the hypothesis that the effect of a prognostic factor on changes after an intermediary event, which cannot be directly tested with the Lunn-McNeil method. Bootstrap-based standard errors improve the coverage rates for Markov model estimates. Overall, the results of our simulations validate Markov multi-state model for a wide range of data structures encountered in prognostic studies of disease progression, and may guide end users regarding the choice of model(s) most appropriate for their specific application.     

Robust minimum distance inference based on combined distances

The minimum disparity estimators proposed by Lindsay (1994) for discrete models form an attractive subclass of minimum distance estimators which achieve their robustness without sacrificing first order efficiency at the model. Similarly, disparity test statistics are useful robust alternatives to the likelihood ratio test for testing of hypotheses in parametric models; they are asymptotically equivalent to the likelihood ratio test statistics under the null hypothesis and contiguous alternatives. Despite their asymptotic optimality properties, the small sample performance of many of the minimum disparity estimators and disparity tests can be considerably worse compared to the maximum likelihood estimator and the likelihood ratio test respectively. In this paper we focus on the class of blended weight Hellinger distances, a general subfamily of disparities, and study the effects of combining two different distances within this class to generate the family of “combined” blended weight Hellinger distances, and identify the members of this family which generally perform well. More generally, we investigate the class of "combined and penal-ized" blended weight Hellinger distances; the penalty is based on reweighting the empty cells, following Harris and Basu (1994). It is shown that some members of the combined and penalized family have rather attractive properties     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.