Testing Inference in Inflated Beta Regressions under Model Misspecification

We consider testing inference in inflated beta regressions subject to model misspecification. In particular, quasi-z tests based on sandwich covariance matrix estimators are described and their finite sample behavior is investigated via Monte Carlo simulations. The numerical evidence shows that quasi-z testing inference can be considerably more accurate than inference made through the usual z tests, especially when there is model misspecification. Interval estimation is also considered. We also present an empirical application that uses real (not simulated) data.     

A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model

The assumption that all random errors in the linear regression model share the same variance (homoskedasticity) is often violated in practice. The ordinary least squares estimator of the vector of regression parameters remains unbiased, consistent and asymptotically normal under unequal error variances. Many practitioners then choose to base their inferences on such an estimator. The usual practice is to couple it with an asymptotically valid estimation of its covariance matrix, and then carry out hypothesis tests that are valid under heteroskedasticity of unknown form. We use numerical integration methods to compute the exact null distributions of some quasi-t test statistics, and propose a new covariance matrix estimator. The numerical results favor testing inference based on the estimator we propose.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

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.     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

Tests of regression coefficients under ridge regression models

This paper investigates two “non-exact” t-type tests, t( k₂) and t(k₂), of the individual coefficients of a linear regression model, based on two ordinary ridge estimators. The reported results are built on a simulation study covering 84 different models. For models with large standard errors, the ridge-based t-tests have correct levels with considerable gain in powers over those of the least squares t-test, t(0). For models with small standard errors, t(k₁) is found to be liberal and is not safe to use while, t(k₂) is found to slightly exceed the nominal level in few cases. When tie two ridge tests art: not winners, the results indicate that they don't loose much against t(0).     

Penalized regression with model‐based penalties

Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function ∫ μ based on observations ti, Yi is the minimizer of Σ{Yi ‐ μ(ti)}² + λ∫(μ′′)². Since ∫(μ″)² is zero when μ is a line, the cubic smoothing spline estimate favors the parametric model μ(t) = α_o + α₁t. Here the authors consider replacing ∫(μ″)² with the more general expression ∫(Lμ)² where L is a linear differential operator with possibly nonconstant coefficients. The resulting estimate of μ performs well, particularly if Lμ is small. They present an O(n) algorithm for the computation of μ. This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study their estimates via simulation and apply them to several data sets.     

Nonparametric two‐step regression estimation when regressors and error are dependent

This paper considers estimation of the function g in the model Y_t = g(X_t ) + ?_t when E(?_t|Xt) ≠ 0 with nonzero probability. We assume the existence of an instrumental variable Z_t that is independent of ?_t, and of an innovation ηt = X_t — E(X_t|Z_t). We use a nonparametric regression of X_t on Z_t to obtain residuals η_t, which in turn are used to obtain a consistent estimator of g. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.     

Mixture of linear mixed models using multivariate t distribution

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off between estimation robustness and efficiency. Simulation studies and an application on analysing lung growth longitudinal data showcase the efficacy of the proposed approach.     

Estimation and hypothesis testing in multivariate linear regression models under non normality

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

Exact unconditional testing procedures for comparing two independent Poisson rates

We consider seven exact unconditional testing procedures for comparing adjusted incidence rates between two groups from a Poisson process. Exact tests are always preferable due to the guarantee of test size in small to medium sample settings. Han [Comparing two independent incidence rates using conditional and unconditional exact tests. Pharm Stat. 2008;7(3):195–201] compared the performance of partial maximization p-values based on the Wald test statistic, the likelihood ratio test statistic, the score test statistic, and the conditional p-value. These four testing procedures do not perform consistently, as the results depend on the choice of test statistics for general alternatives. We consider the approach based on estimation and partial maximization, and compare these to the ones studied by Han (2008) for testing superiority. The procedures are compared with regard to the actual type I error rate and power under various conditions. An example from a biomedical research study is provided to illustrate the testing procedures. The approach based on partial maximization using the score test is recommended due to the comparable performance and computational advantage in large sample settings. Additionally, the approach based on estimation and partial maximization performs consistently for all the three test statistics, and is also recommended for use in practice.     

The split sample permutation t-tests

Without the exchangeability assumption, permutation tests for comparing two population means do not provide exact control of the probability of making a Type I error. Another drawback of permutation tests is that it cannot be used to test hypothesis about one population. In this paper, we propose a new type of permutation tests for testing the difference between two population means: the split sample permutation t-tests. We show that the split sample permutation t-tests do not require the exchangeability assumption, are asymptotically exact and can be easily extended to testing hypothesis about one population. Extensive simulations were carried out to evaluate the performance of two specific split sample permutation t-tests: the split in the middle permutation t-test and the split in the end permutation t-test. The simulation results show that the split in the middle permutation t-test has comparable performance to the permutation test if the population distributions are symmetric and satisfy the exchangeability assumption. Otherwise, the split in the end permutation t-test has significantly more accurate control of level of significance than the split in the middle permutation t-test and other existing permutation tests.     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.     

Variable selection for first‐order Poisson integer‐valued autoregressive model with covariables

In recent years, modelling count data has become one of the most important and popular topics in time‐series analysis. At the same time, variable selection methods have become widely used in many fields as an effective statistical modelling tool. In this paper, we consider using a variable selection method to solve a modelling problem regarding the first‐order Poisson integer‐valued autoregressive (PINAR(1)) model with covariables. The PINAR(1) model with covariables is widely used in many areas because of its practicality. When using this model to deal with practical problems, multiple covariables are added to the model because it is impossible to know in advance which covariables will affect the results. But the inclusion of some insignificant covariables is almost impossible to avoid. Unfortunately, the usual estimation method is not adequate for the task of deleting the insignificant covariables that cause statistical inferences to become biased. To overcome this defect, we propose a penalised conditional least squares (PCLS) method, which can consistently select the true model. The PCLS estimator is also provided and its asymptotic properties are established. Simulation studies demonstrate that the PCLS method is effective for estimation and variable selection. One practical example is also presented to illustrate the practicability of the PCLS method.     

Tests of heteroscedasticity and correlation in multivariate t regression models with AR and ARMA errors

Heteroscedasticity checking in regression analysis plays an important role in modelling. It is of great interest when random errors are correlated, including autocorrelated and partial autocorrelated errors. In this paper, we consider multivariate t linear regression models, and construct the score test for the case of AR(1) errors, and ARMA(s,d) errors. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. Based on modified profile likelihood, the adjusted score test is also developed. The finite sample performance of the tests is investigated through Monte Carlo simulations, and also the tests are illustrated with two real data sets.     

Testing inference in variable dispersion beta regressions

The class of beta regression models proposed by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions, Journal of Applied Statistics 31 (2004), pp. 799–815] is useful for modelling data that assume values in the standard unit interval (0, 1). The dependent variable relates to a linear predictor that includes regressors and unknown parameters through a link function. The model is also indexed by a precision parameter, which is typically taken to be constant for all observations. Some authors have used, however, variable dispersion beta regression models, i.e., models that include a regression submodel for the precision parameter. In this paper, we show how to perform testing inference on the parameters that index the mean submodel without having to model the data precision. This strategy is useful as it is typically harder to model dispersion effects than mean effects. The proposed inference procedure is accurate even under variable dispersion. We present the results of extensive Monte Carlo simulations where our testing strategy is contrasted to that in which the practitioner models the underlying dispersion and then performs testing inference. An empirical application that uses real (not simulated) data is also presented and discussed.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.