On inference for a semiparametric partially linear regression model with serially correlated errors

The authors consider a semiparametric partially linear regression model with serially correlated errors. They propose a new way of estimating the error structure which has the advantage that it does not involve any nonparametric estimation. This allows them to develop an inference procedure consisting of a bandwidth selection method, an efficient semiparametric generalized least squares estimator of the parametric component, a goodness‐of‐fit test based on the bootstrap, and a technique for selecting significant covariates in the parametric component. They assess their approach through simulation studies and illustrate it with a concrete application.     

Pseudo‐likelihood ratio tests for semiparametric multivariate copula model selection

The authors propose pseudo‐likelihood ratio tests for selecting semiparametric multivariate copula models in which the marginal distributions are unspecified, but the copula function is parameterized and can be misspecified. For the comparison of two models, the tests differ depending on whether the two copulas are generalized nonnested or generalized nested. For more than two models, the procedure is built on the reality check test of White (2000). Unlike White (2000), however, the test statistic is automatically standardized for generalized nonnested models (with the benchmark) and ignores generalized nested models asymptotically. The authors illustrate their approach with American insurance claim data.     

Longitudinal data analysis for generalized linear models with follow‐up dependent on outcome‐related variables

In longitudinal studies, observation times are often irregular and subject‐specific. Frequently they are related to the outcome measure or other variables that are associated with the outcome measure but undesirable to condition upon in the model for outcome. Regression analyses that are unadjusted for outcome‐dependent follow‐up then yield biased estimates. The authors propose a class of inverse‐intensity rate‐ratio weighted estimators in generalized linear models that adjust for outcome‐dependent follow‐up. The estimators, based on estimating equations, are very simple and easily computed; they can be used under mixtures of continuous and discrete observation times. The predictors of observation times can be past observed outcomes, cumulative values of outcome‐model covariates and other factors associated with the outcome. The authors validate their approach through simulations and they illustrate it using data from a supported housing program from the US federal government.     

On quasi‐likelihood inference in generalized linear mixed models with two components of dispersion

The authors propose a quasi‐likelihood approach analogous to two‐way analysis of variance for the estimation of the parameters of generalized linear mixed models with two components of dispersion. They discuss both the asymptotic and small‐sample behaviour of their estimators, and illustrate their use with salamander mating data.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Analyzing multivariate longitudinal binary data: A generalized estimating equations approach

The authors consider regression analysis for binary data collected repeatedly over time on members of numerous small clusters of individuals sharing a common random effect that induces dependence among them. They propose a mixed model that can accommodate both these structural and longitudinal dependencies. They estimate the parameters of the model consistently and efficiently using generalized estimating equations. They show through simulations that their approach yields significant gains in mean squared error when estimating the random effects variance and the longitudinal correlations, while providing estimates of the fixed effects that are just as precise as under a generalized penalized quasi‐likelihood approach. Their method is illustrated using smoking prevention data.     

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Enhancements of Non‐parametric Generalized Likelihood Ratio Test: Bias Correction and Dimension Reduction

Non‐parametric generalized likelihood ratio test is a popular method of model checking for regressions. However, there are two issues that may be the barriers for its powerfulness: existing bias term and curse of dimensionality. The purpose of this paper is thus twofold: a bias reduction is suggested and a dimension reduction‐based adaptive‐to‐model enhancement is recommended to promote the power performance. The proposed test statistic still possesses the Wilks phenomenon and behaves like a test with only one covariate. Thus, it converges to its limit at a much faster rate and is much more sensitive to alternative models than the classical non‐parametric generalized likelihood ratio test. As a by‐product, we also prove that the bias‐corrected test is more efficient than the one without bias reduction in the sense that its asymptotic variance is smaller. Simulation studies and a real data analysis are conducted to evaluate of proposed tests.     

Testing noninferiority in three‐armed clinical trials based on likelihood ratio statistics

Clinical noninferiority trials with at least three groups have received much attention recently, perhaps due to the fact that regulatory agencies often require that a placebo group be evaluated along with a new experimental drug and an active control. The authors discuss likelihood ratio tests for binary endpoints and various noninferiority hypotheses. They find that, depending on the particular hypothesis, the test reduces asymptotically either to the intersection‐union test or to a test which follows asymptotically a mixture of generalized chi‐squared distributions. They investigate the performance of this asymptotic test and provide an exact modification. They show that this test considerably outperforms multiple testing methods such as the Bonferroni adjustment with respect to power. They illustrate their methods with a cancer study to compare antiemetic agents. Finally, they discuss the extension of the results to other settings, such as Gaussian endpoints.     

Bayesian analysis of mark‐recapture data with travel time‐dependent survival probabilities

The authors extend the classical Cormack‐Jolly‐Seber mark‐recapture model to account for both temporal and spatial movement through a series of markers (e.g., dams). Survival rates are modeled as a function of (possibly) unobserved travel times. Because of the complex nature of the likelihood, they use a Bayesian approach based on the complete data likelihood, and integrate the posterior through Markov chain Monte Carlo methods. They test the model through simulations and apply it also to actual salmon data arising from the Columbia river system. The methodology was developed for use by the Pacific Ocean Shelf Tracking (POST) project.     

Corrected local polynomial estimation in varying‐coefficient models with measurement errors

The authors study a varying‐coefficient regression model in which some of the covariates are measured with additive errors. They find that the usual local linear estimator (LLE) of the coefficient functions is biased and that the usual correction for attenuation fails to work. They propose a corrected LLE and show that it is consistent and asymptotically normal, and they also construct a consistent estimator for the model error variance. They then extend the generalized likelihood technique to develop a goodness of fit test for the model. They evaluate these various procedures through simulation studies and use them to analyze data from the Framingham Heart Study.     

On testing for multivariate ARCH effects in vector time series models

Using a spectral approach, the authors propose tests to detect multivariate ARCH effects in the residuals from a multivariate regression model. The tests are based on a comparison, via a quadratic norm, between the uniform density and a kernel‐based spectral density estimator of the squared residuals and cross products of residuals. The proposed tests are consistent under an arbitrary fixed alternative. The authors present a new application of the test due to Hosking (1980) which is seen to be a special case of their approach involving the truncated uniform kernel. However, they typically obtain more powerful procedures when using a different weighting. The authors consider especially the procedure of Robinson (1991) for choosing the smoothing parameter of the spectral density estimator. They also introduce a generalized version of the test for ARCH effects due to Ling & Li (1997). They investigate the finite‐sample performance of their tests and compare them to existing tests including those of Ling & Li (1997) and the residual‐based diagnostics of Tse (2002).Finally, they present a financial application.     

Likelihood inference for small variance components

The authors explore likelihood‐based methods for making inferences about the components of variance in a general normal mixed linear model. In particular, they use local asymptotic approximations to construct confidence intervals for the components of variance when the components are close to the boundary of the parameter space. In the process, they explore the question of how to profile the restricted likelihood (REML). Also, they show that general REML estimates are less likely to fall on the boundary of the parameter space than maximum‐likelihood estimates and that the likelihood‐ratio test based on the local asymptotic approximation has higher power than the likelihood‐ratio test based on the usual chi‐squared approximation. They examine the finite‐sample properties of the proposed intervals by means of a simulation study.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

A Model Specification Test For GARCH(1,1) Processes

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér‐von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model‐based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.     

Inference for general parametric functions in box‐Cox‐type transformation models

The authors propose a simple but general method of inference for a parametric function of the Box‐Cox‐type transformation model. Their approach is built upon the classical normal theory but takes parameter estimation into account. It quickly leads to test statistics and confidence intervals for a linear combination of scaled or unsealed regression coefficients, as well as for the survivor function and marginal effects on the median or other quantité functions of an original response. The authors show through simulations that the finite‐sample performance of their method is often superior to the delta method, and that their approach is robust to mild departures from normality of error distributions. They illustrate their approach with a numerical example.     

A Statistical Test of Change‐Point in Mean that Almost Surely Has Zero Error Probabilities

In this paper we develop a non‐conventional statistical test for the change‐point in a mean model by making use of an almost‐sure (a.s.) convergence (or strong convergence) result that we obtain, in respect of the difference between the sums of squared residuals under the null and alternative hypotheses. We prove that both types of error probabilities of the new test converge to zero almost surely when the sample size goes to infinity. This result does not hold for any conventional statistical test where the type I error probability, i.e. the significance level or the size, is prescribed at a low but non‐zero level (e.g. 0.05). The test developed is easy to use in practice, and is ready to be generalised to other change‐point models provided that the relevant almost‐sure convergence results are available. We also provide a simulation study in the paper to compare the new and conventional tests under different data scenarios. The results obtained are consistent with our asymptotic study. In addition we provide least squares estimators of those parameters used in the change‐point test together with their almost‐sure convergence properties.     

Bayesian robust transformation and variable selection: A unified approach

The authors consider the problem of simultaneous transformation and variable selection for linear regression. They propose a fully Bayesian solution to the problem, which allows averaging over all models considered including transformations of the response and predictors. The authors use the Box‐Cox family of transformations to transform the response and each predictor. To deal with the change of scale induced by the transformations, the authors propose to focus on new quantities rather than the estimated regression coefficients. These quantities, referred to as generalized regression coefficients, have a similar interpretation to the usual regression coefficients on the original scale of the data, but do not depend on the transformations. This allows probabilistic statements about the size of the effect associated with each variable, on the original scale of the data. In addition to variable and transformation selection, there is also uncertainty involved in the identification of outliers in regression. Thus, the authors also propose a more robust model to account for such outliers based on a t‐distribution with unknown degrees of freedom. Parameter estimation is carried out using an efficient Markov chain Monte Carlo algorithm, which permits moves around the space of all possible models. Using three real data sets and a simulated study, the authors show that there is considerable uncertainty about variable selection, choice of transformation, and outlier identification, and that there is advantage in dealing with all three simultaneously. The Canadian Journal of Statistics 37: 361–380; 2009 © 2009 Statistical Society of Canada