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Summary The classical integrated conditional moment test is a promising method for model checking and its basic idea has been applied to develop several variants. However, in diverging-dimension scenarios, the integrated conditional moment test may break down and has completely different limiting properties from the fixed-dimension case. Furthermore, the related wild bootstrap approximation can also be invalid. To extend this classical test to diverging dimension settings, we propose a projected adaptive-to-model version of the integrated conditional moment test. We study the asymptotic properties of the new test under both the null and alternative hypotheses to examine if it maintains significance level, and its sensitivity to the global and local alternatives that are distinct from the null at the rate $n^{-1/2}$. The corresponding wild bootstrap approximation can still work for the new test in diverging-dimension scenarios. We also derive the consistency and asymptotically linear representation of the least squares estimator when the parameter diverges at the fastest possible known rate in the literature. Numerical studies show that the new test can greatly enhance the performance of the integrated conditional moment test in high-dimensional cases. We also apply the test to a real dataset for illustration.

We study tools for checking the validity of a parametric regression model. When the dimension of the regressors is large, many of the existing tests face the curse of dimensionality or require some ordering of the data. Our tests are based on the residual empirical process marked by proper functions of the regressors. They are able to detect local alternatives converging to the null at parametric rates. Parametric and nonparametric alternatives are considered. In the latter case, through a proper principal component decomposition, we are able to derive smooth directional tests which are asymptotically distribution-free under the null model. The new tests take into account precisely the 'geometry of the model'. A simulation study is carried through and an application to a real dataset is illustrated. Copyright 2008, Oxford University Press.

A frequently encountered challenge in high-dimensional regression is the detection of relevant variables. Variable selection suffers from instability and the power to detect relevant variables is typically low if predictor variables are highly correlated. When taking the multiplicity of the testing problem into account, the power diminishes even further. To gain power and insight, it can be advantageous to look for influence not at the level of individual variables but rather at the level of clusters of highly correlated variables. We propose a hierarchical approach. Variable importance is first tested at the coarsest level, corresponding to the global null hypothesis. The method then tries to attribute any effect to smaller subclusters or even individual variables. The smallest possible clusters, which still exhibit a significant influence on the response variable, are retained. It is shown that the proposed testing procedure controls the familywise error rate at a prespecified level, simultaneously over all resolution levels. The method has power comparable to the Bonferroni--Holm procedure on the level of individual variables and dramatically larger power for coarser resolution levels. The best resolution level is selected adaptively. Copyright 2008, Oxford University Press.

When some of the records used to estimate the imputation models in multiple imputation are not used or available for analysis, the usual multiple imputation variance estimator has positive bias. We present an alternative approach that enables unbiased estimation of variances and, hence, calibrated inferences in such contexts. First, using all records, the imputer samples m values of the parameters of the imputation model. Second, for each parameter draw, the imputer simulates the missing values for all records n times. From these mn completed datasets, the imputer can analyse or disseminate the appropriate subset of records. We develop methods for interval estimation and significance testing for this approach. Methods are presented in the context of multiple imputation for measurement error. Copyright 2008, Oxford University Press.

We present a new computationally feasible test for the dimension of the central subspace in a regression problem based on sliced average variance estimation. We also provide a marginal coordinate test. Under the null hypothesis, both the test of dimension and the marginal coordinate test involve test statistics that asymptotically have chi-squared distributions given normally distributed predictors, and have a distribution that is a linear combination of chi-squared distributions in general. Copyright 2007, Oxford University Press.

The results of Davies (1977, 1987) are extended to a linear model situation with unknown residual variance.

SUMMARY Generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified. However, under such misspecification, the estimator of the regression parameter can be inefficient. In this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter, and that remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices, a representation that is valid for the working correlations most commonly used. Both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations. This approach also provides a chi-squared inference function for testing nested models and a chi-squared regression misspecification test. Furthermore, the test statistic follows a chi-squared distribution asymptotically whether or not the working correlation structure is correctly specified.

In this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed. The procedure is based on signs of differences of observations from the response variable. The test is calibrated against the most difficult null hypothesis, when the regression function is constant, and produces an exact test in this context. In general, the test is conservative. The power of the test is good, and comparable with that of other nonparametric tests. It is shown that the testing procedure has asymptotic power 1 against certain local alternatives. The method is also robust against heavy-tailed error distributions, and even maintains good power when the errors are for example Cauchy distributed. A simulation study is provided to demonstrate finite-sample behaviour of the testing procedure.

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 Receiver operating characteristic curves (ROC's) are used to evaluate diagnostic tests when test results are not binary. They describe the inherent capacity of the test for distinguishing between truly diseased and nondiseased subjects. Although methodology for estimating and for comparing ROC'S is well developed, to date no general framework exists for evaluating covariate effects on ROC'S. We formulate a general regression model which allows the effects of covariates on test accuracy to be succinctly summarised. Such covariates might include, for example, characteristics of the patient or test environment, test type or severity of disease. The regression models are shown to arise naturally from some classic models for continuous or ordinal test data. Regression parameters are fitted using an estimating equation approach. The method is illustrated on data from a study of multiformat photographic images used for scintigraphy.

SUMMARY This paper is concerned with the null hypothesis that errors in a regression equation for time series data follow a random walk. We examine the power properties of most powerful invariant tests for the unit root null hypotheses against exact stationary and nonstationary first order autoregressive models. The analysis shows the importance of a constant term and a linear trend variable in certain cases. The implications of the results for models estimated using seasonal data are briefly discussed.

SUMMARY In this paper, tests for structural change based on moving sums (MosuMs) of recursive and least-squares residuals are investigated. We obtain and tabulate the asymptotic critical values of the MOSUM test with recursive residuals and show that the asymptotic critical values of the MOSUM test with least-squares residuals can easily be obtained from already existing tables for the moving-estimates test. We also show that these MOSUM tests are consistent and have nontrivial local power against a general class of alternatives. Our simulations further indicate that the proposed MOSUM tests can complement other tests when there is a single structural change and have power advantage when there are certain double structural changes.

SUMMARY Gill & Schumacher (1987) assess proportionality of hazards in the two-sample case by comparing two generalized linear rank tests with different weights. They consider the difference of the two linear rank statistics; under proportionality, the expected value of this difference is zero and its variance may be estimated. The purpose of this note is to show that one can perform a similar procedure within the traditional regression setting. The procedure has the advantage of generalizability to multi-sample or continuous data and is computable using existing regression programs.

SUMMARY Limit distributions both under the null and a sequence of alternative hypotheses are derived for the likelihood ratio and related statistics for testing the homogeneity of several autoregressive processes. These results are extended to mixed autoregressive and moving average processes. It is often required to test whether two or more sets of data can be considered as coming from a common time series model. If the hypothesis of homogeneity is accepted one can obviously achieve parsimony in model specification and also obtain better estimates of the model parameters by pooling the data sets. On the other hand, if there is a significant difference between the sets one may be interested in alternative models which may vary between sets but are still close to each other for large samples. When dealing with regression models the homogeneity tests usually lead to analysis of variance techniques. In this note we consider the comparison of autoregressive and also of mixed autoregressive-moving average models. A unified method of deriving limit distributions of the likelihood ratio and related test statistics is suggested in ? 2. The method is illustrated in ? 3 for autoregressive processes augmented by regression variables, and for mixed autoregressive-moving average processes.

SUMMARY This paper investigates the distribution of the locally best invariant test for coefficient constancy, against the random walk alternative, in a regression model which may contain nonvarying coefficients. Unfortunately, the distribution of the locally best invariant test depends critically on the exogenous variables of the model, even asymptotically. A modification is suggested which has a known asymptotic distribution regardless of the regression effect.

Cox (1972) proposed a regression model for life tables and derived certain hypothesis testing procedures. If there is no censoring and we wish to test a simple null hypothesis, his test statistic is similar to a permutation test. This relation is explored and it is shown how to modify any rank test to allow for censored data. Cox's test and the log rank test are compared and it is concluded that if no censoring occurs the nonparametric test is to be preferred if the sample size is very small. When censoring occurs tests like Cox's are recommended.

A great deal of use has undoubtedly been made of least squares regression methods in circumstances in which they are known to be inapplicable. In particular, they have often been employed for the analysis of time series and similar data in which successive observations are serially correlated. The resulting complications are well known and have recently been studied from the standpoint of the econometrician by Cochrane & Orcutt (1949). A basic assumption underlying the application of the least squares method is that the error terms in the regression model are independent. When this assumption—among others—is satisfied the procedure is valid whether or not the observations themselves are serially correlated. The problem of testing the errors for independence forms the subject of this paper and its successor. The present paper deals mainly with the theory on which the test is based, while the second paper describes the test procedures in detail and gives tables of bounds to the significance points of the test criterion adopted. We shall not be concerned in either paper with the question of what should be done if the test gives an unfavourable result.

In two previous papers, a model for multivariate paired comparisons was proposed and the associated methodology and large sample properties were developed. The present study considers the problem of relating the response pattern on overall quality to that on a specified set of attributes. A regression equation is derived for a joint distribution of responses to dichotomous items and is applied to the multivariate paired comparison model. A test of the significance of the responses to specified attributes in estimating the responses to overall quality is presented. (Author)

There are a number of variable selection procedures in current use in multiple regression analysis. In some of these procedures, a new variable is entered into regression when its contribution to the explanation of the response is larger than that of any other variable not yet entered. The new entrant is then tested by a standard F-test to see if its contribution is 'significant'. Because of the method of selecting the largest contributor, an F-test of this type cannot be theoretically correct but the procedure is used because the exact distributions are not known. This raises questions like 'Is one seriously in error in using the F-tests.' 'If so, can adjustments be made to the F-test to correct the difficulty.' The purpose of this paper is to discuss some new results which throw light on these questions. (Author)

SUMMARY A class of simple estimators for the correlation of the errors in nonparametric regression models is proposed for the fixed design case. These estimators are based on squared differences of various spans of the data and are consistent so long as the regression function is Lipschitz continuous. The limiting distribution is established and applied to derive asymptotic 'global' and 'local' tests for different null hypotheses of uncorrelatedness. The finite sample behaviour of estimators and tests is investigated in a Monte Carlo study. A more general class of estimators using generalized difference schemes is also introduced.