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Summary The estimation of time series models with heavy-tailed innovations has been widely discussed, but corresponding goodness-of-fit tests have attracted less attention, primarily because the autocorrelation function commonly used in constructing goodness-of-fit tests necessarily imposes certain moment conditions on the innovations. As a bounded random variable has finite moments of all orders, we address the problem by first transforming the residuals with a bounded function. More specifically, we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model. With the corresponding residual empirical distribution function naturally employed as the transformation, a robust goodness-of-fit test is then constructed. The asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived, and Monte Carlo experiments are conducted to examine finite-sample properties. The proposed test is shown to be more powerful than existing tests when the innovations are heavy-tailed.

This paper addresses the topic of nonnested time series model comparisons. The main result is a central limit theorem for the likelihood ratio statistic when the models are nonnested and non-equivalent. The concepts of model equivalence and forecast equivalence, which are important for determining the parameter subset corresponding to the null hypothesis, are developed. The method is validated through a simulation study and illustrated on a retail time series.

Coarse structural nested mean models are tools for estimating treatment effects from longitudinal observational data with time-dependent confounding. There is, however, no guidance on how to specify the treatment effect model, and model misspecification can lead to bias. We derive a goodness-of-fit test based on modified over-identification restrictions tests for evaluating a treatment effect model, and show that our test is doubly robust in the sense that, with a correct treatment effect model, the test has the correct Type I error if either the treatment initiation model or a nuisance regression outcome model is correctly specified. In a simulation study, we show that the test has correct Type I error and can detect model misspecification. We use the test to study how the timing of antiretroviral treatment initiation after HIV infection predicts the effect of one year of treatment in HIV-positive patients with acute and early infection.

We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and predictor variables and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the predictor and the residual obtained from the parametric fit by using the Hilbert–Schmidt independence criterion (Gretton et al., 2008). The proposed method requires no user-defined regularization, is simple to compute based on only pairwise distances between points in the sample, and is consistent against all alternatives. We develop distribution theory for the proposed test statistic, under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap scheme. A simulation study shows that our method has better power than its main competitors. Two real datasets are analysed to demonstrate the scope and usefulness of our method.

We investigate diagnostic measures for assessing the influence of observations and model misspecification on the Cox regression model when there are missing covariate data. Our diagnostics include case-deletion measures, conditional martingale residuals, and score residuals. The Q-distance is introduced to examine the effects of deleting individual observations on the estimates of finite- and infinite-dimensional parameters. Conditional martingale residuals are used to construct goodness-of-fit statistics for testing misspecification of the model assumptions. A resampling method is developed to approximate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$p$\end{document}-values of the goodness-of-fit statistics. We conduct simulation studies to evaluate our methods, and analyse a real dataset to illustrate their use.

We study goodness-of-fit tests for logistic regression models for case-control data when some covariates are measured with error. We first study the applicability of traditional test methods for this problem, simply ignoring measurement error, and show that in some scenarios they are effective despite the inconsistency of the parameter estimators. We then develop a test procedure based on work of Zhang (2001) that can simultaneously test the validity of logistic regression and correct the bias in parameter estimators for case-control data with nondifferential classical additive normal measurement error. Instead of using the information matrix considered by Zhang (2001), our test statistic uses preselected functions to reduce dimensionality. Simulation studies and an application illustrate its usefulness. Copyright 2011, Oxford University Press.

This paper discusses copula model selection procedures and goodness-of-fit tests under censoring. The proposed methodology is based on a comparison of nonparametric and model-based estimators of the probability integral transformation, K. New weighted estimators for K are introduced. The resulting tests are compared to an existing approach by simulation and illustrated with an example involving bleeding changes in a woman's reproductive history. Copyright 2010, Oxford University Press.

In the generalized method of moments approach to longitudinal data analysis, unbiased estimating functions can be constructed to incorporate both the marginal mean and the correlation structure of the data. Increasing the number of parameters in the correlation structure corresponds to increasing the number of estimating functions. Thus, building a correlation model is equivalent to selecting estimating functions. This paper proposes a chi-squared test to choose informative unbiased estimating functions. We show that this methodology is useful for identifying which source of correlation it is important to incorporate when there are multiple possible sources of correlation. This method can also be applied to determine the optimal working correlation for the generalized estimating equation approach. Copyright 2008, Oxford University Press.

We develop diagnostic measures for assessing the influence of individual observations when using empirical likelihood with general estimating equations, and we use these measures to construct goodness-of-fit statistics for testing possible misspecification in the estimating equations. Our diagnostics include case-deletion measures, local influence measures and pseudo-residuals. Our goodness-of-fit statistics include the sum of local influence measures and the processes of pseudo-residuals. Simulation studies are conducted to evaluate our methods, and real datasets are analyzed to illustrate the use of our diagnostic measures and goodness-of-fit statistics. Copyright 2008, Oxford University Press.

We introduce a formal testing procedure to assess the fit of an inhomogeneous spatial Poisson process model, based on a discrepancy measure function that is constructed from residuals obtained from the fitted model. We derive the asymptotic distributional properties of and develop a test statistic based on them. Our test statistic has a limiting standard normal distribution, so that the test can be performed by simply comparing the test statistic with readily available critical values. We perform a simulation study to assess the performance of the proposed method and apply it to a real data example. Copyright 2008, Oxford University Press.

We present a goodness-of-fit test for the logistic regression model under case-control sampling. The test statistic is constructed via a discrepancy between two competing kernel density estimators of the underlying conditional distributions given case-control status. The proposed goodness-of-fit test is shown to compare very favourably with previously proposed tests for case-control sampling in terms of power. The test statistic can be easily computed as a quadratic form in the residuals from a prospective logistic regression maximum likelihood fit. In addition, the proposed test is affine invariant and has an alternative representation in terms of empirical characteristic functions. Copyright 2007, Oxford University Press.

In many applications, the outcome of interest is a mark such that its observation is contingent upon occurrence of an event. With incomplete follow-up data, the marginal mark distribution is, however, nonparametrically nowhere identifiable in many practical situations. To address this problem, we suggest a semiparametric model that postulates a normal copula for the association between the mark and survival time, but leaves the marginals unspecified. We show identifiability of the marginal mark distribution under this model, and propose an inference procedure. The estimated marginal distribution function is consistent and asymptotically normal, and it provides a basis for estimating summaries of the mark. Furthermore, we propose graphical model-checking methods and Kolmogorov-Smirnov-type goodness-of-fit tests. Simulation studies demonstrate that the inference procedure performs well in practical settings. The method is applied to the estimation of lifetime medical cost in a lung cancer trial.

SUMMARY Interval-censored lifetime data arise when individuals in a study are inspected inter mittently so that a lifetime is observed to lie between two successive times. In settings where only these two times are available, methods exist for nonparametric or parametric estimation of lifetime distributions. However, there has been virtually no discussion of how inspection processes may be estimated or identified. Such estimates are needed if one is to generate interval-censored data by simulation. This paper identifies which aspects of an independent inspection process are estimable from interval-censored data, and shows how to obtain nonparametric estimates. The results allow interval-censored data from any specified distribution to be generated, and give new simulation procedures for estimation or testing. A new omnibus goodness-of-fit test is introduced.

SUMMARY The problem of testing for the equality of k distribution functions under Case 2 interval censoring is studied and a supremum-type test statistic is proposed based on the differences between the nonparametric maximum likelihood estimator and the so-called leveraged bootstrap estimator of the k underlying distributions. The proposed test is distribution free and consistent against all alternatives. As the main results hold for a wide range of resampling sizes, a data-driven method is suggested for determining the size of each leveraged bootstrap sample. Another advantage of the test is that it can detect different distributions with equal means or heavy crossover. Simulation studies indicate that the test performs quite well with a moderate sample size. Finally, a slightly modified version of the test is applied to breast cosmesis data.

SUMMARY For bivariate current status data with univariate monitoring times, the identifiable part of the joint distribution is three univariate cumulative distribution functions, namely the two marginal distributions and the bivariate cumulative distribution function evaluated on the diagonal. We show that smooth functionals of these univariate cumulative distribution functions can be efficiently estimated with easily computed nonparametric maximum likelihood estimators based on reduced data consisting of univariate current status observations. This theory is then applied to functionals that address independence of the two survival times and the goodness-of-fit of a copula model used by Wang & Ding (2000). Some brief simulations are provided along with an illustration based on data on HIV transmission. Extension of the ideas to incorporate covariates, possibly time-dependent, are discussed.

SUMMARY Extending the work of Dupuis (1995), we motivate a range of biologically plausible models for multiple-site capture-recapture and show how the original Gibbs sampling algorithm of Dupuis can be extended to obtain posterior model probabilities using reversible jump Markov chain Monte Carlo. This model selection procedure improves upon previous analyses in two distinct ways. First, Bayesian model averaging provides a robust parameter estimation technique which properly incorporates model uncertainty in the resulting intervals. Secondly, by discriminating among perhaps millions of competing models, we are able to discern fine structure within the data and thereby answer questions of primary biological importance. We demonstrate how reversible jump Markov chain Monte Carlo methods provide the only viable method for exploring model spaces of this size. We examine the lizard data discussed in Dupuis (1995) and show that most of the posterior mass is placed upon models not previously considered for these data. We discuss model discrimination and model averaging and focus upon the increased scientific understanding of the data obtained via the Bayesian model comparison procedure.

SUMMARY We address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches. This involves testing for whether or not the zero-mean property of estimating equations holds true. Chen & Little (1999) proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data, whereas we propose a more unified generalised score-type test. This avoids exhaustive estimation of parameters for each missing-data pattern, testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern-specific estimating equations are zero. Comparisons are made for the two approaches with both simulations and real data examples.

We give several definitions of residual autocorrelations and derive their joint asymptotic distribution for the panel time series model of Hjellvik & Tjostheim (1999a). A portmanteau goodness-of-fit test arises naturally from the asymptotic distribution. Simulation results show that the asymptotic standard errors compared satisfactorily with the empirical standard errors, that the goodness-of-fit test has reasonable empirical size, and that it is powerful enough to be useful with a modest sample size. The results of this paper are illustrated with a real-data example. © 2002 Biometrika Trust.

A goodness-of-fit test statistic for spatial point processes is proposed and shown to have an asymptotic chi-squared distribution if the underlying point process is Poisson. Simulations demonstrate that the test, when testing for complete spatial randomness, is more sensitive to mixtures of regular and clustered point processes than the tests using the nearest neighbour distance distribution, the second- or third-order characteristics.

Model misspecification is discussed for the analysis of censored survival data with the Cox regression model. The maximum partial likelihood estimator for covariate effects has an asymptotic normal distribution under model misspecification (Lin & Wei, 1989). Situations where the assumed functional form for covariates is wrong are considered in a general framework in which we also derive the limit of Breslow's (1972) estimator. Furthermore, we give explicit expressions for the asymptotic variance of the maximum partial likelihood estimator. Numerical values for the errors of the maximum partial likelihood estimator and of the usual variance estimator are obtained for functional misspecification of a single relevant covariate and looking at the effect of a first factor, representing treatment or something similar, for functional wrong incorporation of auxiliary covariate measurements. In the latter situations we analyse the impact of dependence between the covariates.

SUMMARY We propose an extension of the classical complete spatial randomness tests to nonstationary Poisson spatial processes. The method consists of first performing the classical tests locally and then grouping the local results into a global test. The global test is a test for nonstationary Poisson process assumption, whereas the local tests can be used in an exploratory way to decide whether the observed process is locally regular or clustered or if we do not reject the inhomogeneous Poisson assumption. Under a Cox assumption, an optimal partition of the sampling window can be derived. Finally, we present some examples from forestry and weed sciences.

SUMMARY We study lack-of-fit tests based on orthogonal series estimators. A common feature of these tests is that they are functions of score statistics that employ data-driven model dimensions. The criteria used to select the dimension are score-based versions of AIC and BIC. The tests can be applied in a wide variety of settings, including both continuous and discrete data. With two or more covariates, a model sequence, i.e. a path in the alternative models space, has to be chosen. Critical points and p-values of the lack-of-fit tests can be obtained via asymptotic distribution theory or by use of the bootstrap. Data examples and a simulation study illustrate the applicability of the tests.

SUMMARY This paper is concerned with simple methods for picking the correct root of a general estimating equation in the case of multiple roots. The use of methods based on (i) asymptotics, (ii) analogues of empirical information and (iii) goodness-of-fit type criteria are advocated, and examples are provided to illustrate their use. It is argued that there is no need to introduce approximate likelihood-ratio or deviance-based methods.

SUMMARY The method of empirical likelihood can be viewed as one of allocating probabilities to an n-cell contingency table so as to minimise a goodness-of-fit criterion. It is shown that, when the Cressie-Read power-divergence statistic is used as the criterion, confidence regions enjoying the same convergence rates as those found for empirical likelihood can be obtained for the entire range of values of the Cressie-Read parameter ,{, including -1, maximum entropy, 0, empirical likelihood, and 1, Pearson's Z2. It is noted that, in the power-divergence family, empirical likelihood is the only member which is Bartlettcorrectable. However, simulation results suggest that, for the mean, using a scaled F distribution yields more accurate coverage levels for moderate sample sizes.

SUMMARY The additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates. We propose two different test procedures for checking the adequacy of two-sample additive risk models for randomly censored observations. One is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk. The test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications. Finally, two real examples are provided, along with results of a simulation study.

SUMMARY Model choice is a fundamental and much discussed activity in the analysis of datasets. Nonnested hierarchical models introducing random effects may not be handled by classical methods. Bayesian approaches using predictive distributions can be used though the formal solution, which includes Bayes factors as a special case, can be criticised. We propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values. We obtain this criterion by minimising posterior loss for a given model and then, for models under consideration, selecting the one which minimises this criterion. For a broad range of losses, the criterion emerges as a form partitioned into a goodness-of-fit term and a penalty term. We illustrate its performance with an application to a large dataset involving residential property transactions.

SUMMARY We propose a simple test of constant conditional hazard ratio in the Clayton model (1978) by comparing the unweighted and weighted concordance estimators of the association parameter. If the Clayton model holds, the difference of these two estimates converges to zero. The proposed test is consistent against alternatives under which the two concordance estimators converge to different values. We derive an explicit formula for the asymptotic variance and derive the asymptotic distribution of the test statistic under the Clayton model. Then we extend the test statistic to incorporate censoring. The proposed test is expected to perform well for a general class of alternatives with monotone conditional hazard ratio. We examine the finite sample properties of the test statistic through simulations.

SUMMARY This paper examines goodness-of-fit testing in the Cox regression model with timevarying regression coefficients. Arjas' (1988) approach is used to define test statistics of the Kolmogorov-Smirnov and Cramer-von Mises types. Their asymptotic limits are shown to be well-known functionals of standard Brownian motion, leading to the construction of formal goodness-of-fit tests. Some numerical studies are also included to illustrate the performance of the tests for moderate sample sizes.

In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector. We then apply the selection rule to specific problems, including testing for pure white noise. The power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression.

SUMMARY Kolmogorov-Smirnov and Cramer-von Mises type test statistics based on the standardised cumulative hazard process are proposed. It is very difficult to evaluate their asymptotic distributions, but they can be approximated by the use of the bootstrap. The advantages of the goodness-of-fit test are that arbitrary partitions of the time axis and covariate spaces are not needed for evaluating test statistics and that it has excellent consistency properties. The test is applied to data from the Mayo Clinic trial in primary biliary cirrhosis of the liver. A simulation study indicates that the proposed test is suitable for practical use.

SUMMARY We test the logistic regression assumption under a case-control sampling plan. After reparameterisation, the assumed logistic regression model is equivalent to a two-sample semiparametric model in which the log ratio of two density functions is linear in data. By identifying this model with a biased sampling model, we propose a Kolmogorov-Smirnovtype statistic to test the validity of the logistic link function. Moreover, we point out that this test statistic can also be used in mixture sampling. We present a bootstrap procedure along with some results on simulation and on analysis of two real datasets.

This paper considers the prediction of a γ-angle, given an x-angle in a circular-circular regression. The predictor under study is a rotation of the decentred x-angle, defined as the direction of the sum of the unit vector corresponding to angle x plus a decentring vector. The parameters are defined as those which maximise the average residual cosine. This predictor is shown to perform well when the data come from a bivariate von Mises distribution or from a wrapped bivariate normal model. The large sample distributions of maximum cosine estimators are derived. Tests of fit for the rotational model, which predicts γ by a rotation of x, and for the independence model where γ is not related to x are presented. The proposed methods are illustrated by the analysis of an earthquake dataset where the direction of ground movement is regressed on the direction of steepest descent.

SUMMARY A general method for doing Monte Carlo simulations conditioned on sufficient statistics is presented. The basic idea is to adjust the parameter values in the corresponding unconditional simulation so that the actual value of the sufficient statistic is obtained. In the case of multiple solutions to this problem, the method has to be modified, even if the corresponding value of the simulated variable is unique for each simulation. The methods are illustrated for some simple models in which the conditional distributions are well known. As a more complicated example, an exact 'analysis of variance' test in the gamma model is performed. Other examples are a modification of the Kolmogorov goodness-of-fit test into an exact test, and minimum variance unbiased estimation of cumulative distributions.

SUMMARY Correlated failure time data arise frequently in scientific investigations because there exists natural or artificial clustering of study subjects such that failure times within the same cluster are correlated. It is convenient and useful to perform regression analysis by formulating the marginal distributions of the correlated failure times with the Cox proportional hazards model. In this paper, we develop a class of graphical and numerical techniques for checking the adequacy of the marginal Cox model. The proposed methods are derived from cumulative sums of martingale-based residuals over the failure time and/or covariates. The distributions of these stochastic processes under the assumed model can be approximated through simulating certain zero-mean Gaussian processes. Each observed residual pattern can then be compared objectively with a number of realisations from the approximating process. An illustration with the Diabetic Retinopathy Study is provided.

SUMMARY This note proposes a test of goodness of fit for time series models based on the sum of the squared residual partial autocorrelations. The test statistic is asymptotically x2. Its small-sample performance is studied through a Monte Carlo experiment. It appears sensitive to erroneous specifications especially when the fitted model understates the order of the moving average component. Residual analysis is a fundamental step in building empirical time series models. When checking the adequacy of the model one usually tests for the absence of residual autocorrelation. There is a number of tests designed for this purpose both in the time and frequency domains; see, for instance, Quenouille (1947, 1949), Bartlett (1954), Box & Pierce (1970), Ljung & Box (1978), Ansley & Newbold (1979), Godfrey (1979). In this paper a test based on residual partial autocorrelations is proposed which is particularly powerful in cases when the fitted model underestimates the order of the moving average component. Let X, be a zero mean process generated by a ARMA model O(B)X, = 0(B)at, where B is the backshift operator, #)(B) is a polynomial of order p, 0(B) is a polynomial of order q and a, is a white noise. Let a', .. , atn be the residuals obtained after estimating the model. A very popular goodness-of-fit test, proposed by Box & Pierce (1970) and improved by Ljung & Box (1978), is based on the statistic m

SUMMARY A non-Gaussian time series model useful for describing the dynamics of endocrinological data is introduced by using a state-space representation. This model is applicable even when the data are sparsely sampled in comparison with time lengths of pulses. Likelihood-based methods of inference for the model are developed. By fitting the present model to partially observed hormonal time series data, we can estimate the locations and heights of the peaks of the pulses as well as their onset times. Models with different shapes of pulses are considered for a comparison of the goodness of fit.

SUMMARY Recent papers by Munson & Jernigan (1989) and Buckley (1991) propose nonparametric tests for the hypothesis of no predictor effect in regression. The Munson-Jernigan test is similar to the von Neumann (1941) test, while that of Buckley is based on a functional of cusums. These tests are shown to be special cases of a wider class of tests based on nonparametric function estimation ideas. Fourier analysis is used to qualitatively compare the Munson & Jernigan and Buckley tests with two new tests constructed from nonparametric smoothers. Their relative powers are then studied by means of large-sample analysis and simulation. The cusum test is the most powerful for very smooth departures from the no-effect hypothesis, while the new tests based on smoothing ideas are clearly superior when the alternative is high frequency.

SUMMARY Binary data in which successive outcomes may be dependent frequently arise in a medical and social sciences context. A simple model for such data is a homogeneous 2-state Markov chain. Under this model one would usually treat the first observation in each sequence, Y1, as fixed. In the case of panel data this may constitute a substantial loss of information. In this paper we propose goodness-of-fit tests for the 2-state Markov model under several likely alternative hypotheses of interest. We also discuss the possibility of assigning to Y, the equilibrium distribution of the chain and outline the gains in efficiency that this can achieve for panel data.

SUMMARY This paper presents a new class of graphical and numerical methods for checking the adequacy of the Cox regression model. The procedures are derived from cumulative sums of martingale-based residuals over follow-up time and/or covariate values. The distributions of these stochastic processes under the assumed model can be approximated by zero-mean Gaussian processes. Each observed process can then be compared, both visually and analytically, with a number of simulated realizations from the approximate null distribution. These comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are. Special attention is given to checking the functional form of a covariate, the form of the link function, and the validity of the proportional hazards assumption. An omnibus test, consistent against any model misspecification, is also studied. The proposed techniques are illustrated with two real data sets.

SUMMARY Rao's score statistic is a standard tool for carrying out hypothesis tests. A unified approach is given which encompasses many common tests. When a suitable group structure is present, it is shown how the derivation of the test statistic can be simplified. An omnibus test of multivariate normality is derived as an example of group invariance. As another application of Rao score tests, a general method is given for constructing tests of independence.

SUMMARY This paper considers the multilevel analysis of structural equation models with unbalanced sampling designs. The analysis is based on the maximum likelihood and the generalized least squares approaches. Basic statistical results for inference such as the asymptotic distribution of the estimators, goodness-of-fit test statistics for the validity of the model and functional constraints are developed. Computationally, the applications of the scoring algorithm and the Gauss-Newton algorithm are discussed.