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The aim of this article is to develop optimal sufficient dimension reduction methodology for the conditional mean in multivariate regression. The context is roughly the same as that of a related method by Cook & Setodji (2003), but the new method has several advantages. It is asymptotically optimal in the sense described herein and its test statistic for dimension always has a chi-squared distribution asymptotically under the null hypothesis. Additionally, the optimal method allows tests of predictor effects. A comparison of the two methods is provided. Copyright 2007, Oxford University Press.

We consider that observations come from a general normal linear model and that it is desirable to test a simplifying null hypothesis about the parameters. We approach this problem from an objective Bayesian, model-selection perspective. Crucial ingredients for this approach are 'proper objective priors' to be used for deriving the Bayes factors. Jeffreys-Zellner-Siow priors have good properties for testing null hypotheses defined by specific values of the parameters in full-rank linear models. We extend these priors to deal with general hypotheses in general linear models, not necessarily of full rank. The resulting priors, which we call 'conventional priors', are expressed as a generalization of recently introduced 'partially informative distributions'. The corresponding Bayes factors are fully automatic, easily computed and very reasonable. The methodology is illustrated for the change-point problem and the equality of treatments effects problem. We compare the conventional priors derived for these problems with other objective Bayesian proposals like the intrinsic priors. It is concluded that both priors behave similarly although interesting subtle differences arise. We adapt the conventional priors to deal with nonnested model selection as well as multiple-model comparison. Finally, we briefly address a generalization of conventional priors to nonnormal scenarios. Copyright 2007, 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.

SUMMARY This paper presents a simple method of calculating one-sided p-values which yield accurate inferences for a simple null hypothesis or, equivalently, accurate one-sided con fidence limits regarding a scalar parameter in the presence of nuisance parameters. The method, which extends the work of DiCiccio et al. (2001) and Lee & Young (2005) in the context of adjusted likelihood estimation, is based on a test statistic analogous to the signed root of the loglikelihood, but derived from the objective function of an M-estimator from a certain class. Monte Carlo simulation is used to avoid the need for onerous analytical calculations typical of competing procedures based on Edgeworth or saddle point approximations. The specific class of M-estimators under consideration is character ised by the requirement that the associated test statistic has constant variance to second order. This class contains, among others, the maximum likelihood estimator as well as variants of commonly used M-estimators of location, such as Huber's (1964) Proposal 2.

We provide a new two-stage procedure in which the linear step-up procedure is used in stage one to estimate mo, providing a new level q' which is used in the linear step-up procedure in the second stage. We prove that a general form of the two-stage procedure controls the false discovery rate at the desired level q. This framework enables us to study analytically the properties of other procedures that exist in the literature. A simulation study is presented that shows that two-stage adaptive procedures improve in power over the original procedure, mainly because they provide tighter control of the false discovery rate. We further study the performance of the current suggestions, some variations of the procedures, and previous suggestions, in the case where the test statistics are positively dependent, a case for which the original procedure controls the false discovery rate. In the setting studied here the newly proposed two-stage procedure is the only one that controls the false discovery rate. The procedures are illustrated with two examples of biological importance.

In this paper we evaluate the characteristics observed both on a crime sample and on individuals included in a database to assess the probability of alternative hypotheses concerning identification. The problem is first addressed by considering a generic characteristic and we demonstrate the problem via a computationally efficient Bayesian network. Then we turn our attention to a heritable DNA trait to show how to evaluate the hypotheses that some individuals, genetically related to the members of the database, are the donors of the crime sample. Then the network is extended to cope with many loci. Applications of the method are provided as well as details of computational requirements.

This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias.

SUMMARY In this note we give an alternative formulation of the nonparametric estimators of Wang (2003) with the identity link. This results in a closed form of the estimator that has computational advantages and gives insight into the rationale behind the estimator.

SUMMARY Popular methods for estimating the central subspace in regression require slicing a continuous response. However, slicing can result in loss of information and in some cases that loss can be substantial. We use intraslice covariances to construct improved inference methods for the central subspace. These methods are optimal within a class of quadratic inference functions and permit chi-squared tests of conditional independence hypotheses involving the predictors. Our experience gained through simulation is that the new method is never worse than existing methods, and can be substantially better.

SUMMARY Typical alternative hypotheses in the analysis of residuals of a standard regression model are considered, and for each one a Bayesian diagnostic based on a symmetric form of the Kullback-Leibler divergence is determined. The results include an explicit expression for the diagnostic when the alternative hypothesis is that the errors are generated by an unknown distribution function with a Dirichlet process prior. This expression is immediately interpretable, exactly computable and endowed with important asymptotic connections. A linear approximation of the diagnostic reveals close links with the class of Lagrange multiplier test statistics. When the alternative hypothesis is that the errors are generated by an autoregressive process the linear approximation is proportional to the Box-Pierce statistic or to the Ljung-Box statistic, according to the characteristics of the prior, if the observations have zero mean; it depends on the Durbin-Watson statistic if the errors are first-order autoregressive, and it is related to the Cliff-Ord statistic if they are generated by a first-order spatial autoregression. The sensitivity to the prior of the diagnostic and of its linear approximation is also discussed.

We derive nonparametric procedures for testing for and against likelihood ratio ordering in the two-population setting with continuous distributions. We account for this ordering by examining the least concave majorant of the ordinal dominance curve formed from the nonparametric maximum likelihood estimators of the continuous distribution functions F and G. In particular, we focus on testing equality of F and G versus likelihood ratio ordering and testing for a violation of likelihood ratio ordering. For both testing problems, we propose area-based and sup-norm-based test statistics, derive appropriate limiting distributions, and provide simulation results that characterise the performance of our procedures. We illustrate our methods using data from a controlled experiment involving the effects of radiation on mice.

SUMMARY In this paper we propose inference methods based on the EM algorithm for estimating the parameters of a weakly parameterised competing risks model with masked causes of failure and second-stage data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause-specific hazard functions and of the masking probabilities is performed via an EM algorithm. Both the E- and M-steps can be solved in closed form under the full model and under some restricted models of interest. We illustrate the flexibility of the method by showing how grouped data and tests of common hypotheses in the literature on missing cause of death can be handled. The method is applied to a real dataset and the asymptotic and robustness properties of the estimators are investigated through simulation.

SUMMARY We propose a method for finding the alternative distribution in importance sampling. The alternative distribution is optimal in the sense that the asymptotic variance is mini mised for estimating tail probabilities of asymptotically normal statistics. Our contribution to importance sampling is three-fold. To begin with, we obtain an explicit expression for the mean of the optimal alternative distribution and the expression motivates a recursive approximation algorithm. Secondly, a new multi-dimensional exponential tilting formula is presented. Lastly, a conservative estimator of the variance is given to facilitate a quick comparison among different stratified sampling schemes in conjunction with importance sampling. Several numerical examples illustrating the efficacy of the proposed method are also included. These results indicate that the proposed method is considerably more efficient than the method based on large deviations theory and the efficiency gain is more significant in higher dimensions.

There are few techniques available for testing whether or not a family of parametric times series models fits a set of data reasonably well without serious restrictions on the forms of alternative models. In this paper, we consider generalised likelihood ratio tests of whether or not the spectral density function of a stationary time series admits certain parametric forms. We propose a bias correction method for the generalised likelihood ratio test of Fan et al. (2001). In particular, our methods can be applied to test whether or not a residual series is white noise. Sampling properties of the proposed tests are established. A bootstrap approach is proposed for estimating the null distribution of the test statistics. Simulation studies investigate the accuracy of the proposed bootstrap estimate and compare the power of the various ways of constructing the generalised likelihood ratio tests as well as some classic methods like the Cramer-von Mises and Ljung-Box tests. Our results favour the newly proposed bias reduction method using the local likelihood estimator.

We propose a test for multimodality with dependent data by resampling from a suitably constructed transition probability kernel, which includes Silverman's test with independent data as a special case. We extend some theoretical properties of Silverman's test with independent and identically distributed data to weakly dependent data, and also discuss the robustness of Silverman's test against departure from independence.

SUMMARY We consider the testing and estimation of thresholds in heteroscedastic threshold autoregressive models with an unknown number of thresholds. A test statistic based on empirical wavelet coefficients is proposed. The asymptotic distribution of the test statistic is established and consistent estimators of the thresholds and the number of thresholds are given. A Monte Carlo study and a real example are used to assess the performance of our method.

Interest in transform methods for normalising test statistics declined with the advent of computers. More recently, small-sample asymptotic methods have been developed to approximate the distributions of complicated test statistics. We propose a generalisation of a classical symmetrising transform to a similar range of statistics. It is shown to behave comparably in a parametric neighbourhood to methods that exploit exponential tilting.

SUMMARY We propose a Lagrange multiplier test for additivity based on the Btirmann expansion of a conditional mean function. The asymptotic null distribution of the test is shown to be x2, under some regularity conditions. In contrast, the Lagrange multiplier test proposed by Chen et al. (1995) is based on the Volterra expansion of the conditional mean function. We discuss some desirable advantages of the Bt*rmann expansion over the Volterra expansion for nonlinear time series modelling. We also reported an empirical study which shows that, in terms of empirical power, the Lagrange multiplier test motivated by the Biirmann expansion outperforms the test of Chen et al. (1995) for the cases for which the Lagrange multiplier test is designed. For other cases for which none of the tests is specifically designed, the empirical powers of the two tests are comparable. Finally, we illustrated the use of the Lagrange multiplier test with a blowfly experimental system.

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.

SUMMARY We derive the limiting distribution of the partial likelihood ratio under general conditions. The multiplicative hazards models being fitted may be nonnested and misspecified. The true model is not assumed to contain either model under consideration. The null hypothesis is that the models are equidistant in Kullback-Leibler metric applied to the rank likelihood. The statistic is consistent for the model which is closer to the truth. Its distribution depends on the unknown data-generating mechanism. A sequential testing procedure is proposed for nonnested comparisons which is valid regardless of the true model. This involves a novel statistic for the equality of the fitted models which is separate from the partial likelihood. The methodology has important applications in model assessment. Simulations and a real example demonstrate its utility in selecting the functional forms of covariates and relative risks.

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.

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

Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H-0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances.

SUMMARY A global smoothing procedure is developed using basis function approximations for estimating the parameters of a varying-coefficient model with repeated measurements. Inference procedures based on a resampling subject bootstrap are proposed to construct confidence regions and to perform hypothesis testing. Conditional biases and variances of our estimators and their asymptotic consistency are developed explicitly. Finite sample properties of our procedures are investigated through a simulation study. Application of the proposed approach is demonstrated through an example in epidemiology. In contrast to the existing methods, this approach applies whether or not the covariates are timeinvariant and does not require binning of the data when observations are sparse at distinct observation times.

SUMMARY We propose a method for comparing survival distributions when cause-of-failure information is missing for some individuals. We use multiple imputation to impute missing causes of failure, where the probability that a missing cause is that of interest may depend on auxiliary covariates, and combine log-rank statistics computed from several 'completed' datasets into a test statistic that achieves asymptotically the nominal level. Simulations demonstrate the relevance of the theory in finite samples.

SUMMARY We have implemented a Lagrange multiplier test for the alternative hypothesis of a nonlinear continuous-time autoregressive model with the instantaneous mean having multiple degrees of nonlinearity. This test is an extension of a Lagrange multiplier test proposed by Tsai & Chan (2000), with the alternative model analogous to the model used in Tsay's (1986) discrete-time work. The performance of the test in the finite-sample case is compared with several existing tests for nonlinearity including Keenan's (1985) test, Petruccelli & Davies' (1986) test, Tsay's (1986, 1989) tests and Tsai & Chan's (2000) test. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregressive models, bilinear models and the nonlinear continuous-time autoregressive models for which the Lagrange multiplier test is designed. In general, the test is more powerful than all the other tests. The test is further illustrated with the annual sunspot data and the lynx data.

SUMMARY We extend the optimal symmetric group sequential tests of Eales & Jennison (1992) to the broader class of asymmetric designs. Two forms of asymmetry are considered, involving unequal type I and type II error rates and different emphases on expected sample sizes at the null and alternative hypotheses. We discuss the properties of our optimal designs and use them to assess the efficiency of the family of tests proposed by Pampallona & Tsiatis (1994) and two families of one-sided tests defined through error spending functions. We show that the error spending designs are highly efficient, while the easily implemented tests of Pampallona & Tsiatis are a little less efficient but still not far from optimal. Our results demonstrate that asymmetric designs can decrease the expected sample size under one hypothesis, but only at the expense of a significantly larger expected sample size under the other hypothesis.

SUMMARY In many tumour growth inhibition studies, tumour sizes are recorded over a period of time, forming a growth curve for each experimental animal. A frequently asked question in these studies is whether or not different treatment agents/inhibitors result in different growth rates. To compare tumour growth rates under two treatments, either the t-test or the Wilcoxon-Mann-Whitney test could be used under suitable assumptions, provided that a one-dimensional variable, such as the area under the curve, is used to summarise a growth curve. Such tests may not be valid in the presence of informative heterogeneous censoring, in that a subject's continued participation in the experiment may depend on the tumour size and/or the level of toxicity of the treatment given. We propose test statistics that naturally correct the bias caused by the censorship and retain high efficiency. They are easy to construct and are nonparametric in nature, making no assumption on the distributions of the growth curves. The method is illustrated with an example from a tumour growth inhibition study on mice. Simulation results are also reported showing that the method performs well with moderate sample sizes.

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 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.

SUMMARY This paper generalises four types of disturbance commonly used in univariate time series analysis to the multivariate case, highlights the differences between univariate and multivariate outliers, and investigates dynamic effects of a multivariate outlier on individual components. The effect of a multivariate outlier depends not only on its size and the underlying model, but also on the interaction between the size and the dynamic structure of the model. The latter factor does not appear in the univariate case. A multivariate outlier can introduce various types of outlier for the marginal component models. By comparing and contrasting results of univariate and multivariate outlier detections, one can gain insights into the characteristics of an outlier. We use real examples to demonstrate the proposed analysis.

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.

A Bayesian analysis for factorial experiments is presented, using finite mixture distributions to model the main effects and interactions. This allows both estimation and an analogue of hypothesis testing in a posterior analysis using a single prior specification. A detailed formulation based on this approach is provided for the case of the two-way model with replication, allowing interactions. Issues in formulating a suitable prior are discussed in detail, and, in the context of two illustrative applications, we discuss implementation, presentation of posterior distributions, sensitivity and performance of the Markov chain Monte Carlo methods that are used.

Summary A simple, nonparametric two-sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions, including the Kaplan‐Meier estimator, a self-consistent estimator for doubly-censored data and an estimator for repeated measures data. The null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diVer between the two samples. This procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry. A simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable.

SUMMARY The often conservative nature, for discrete data, of so-called exact tests seems usually the result of unnecessarily precise conditioning. We consider avoiding this by conditioning only approximately on the sufficient statistics for nuisance parameters. Modest relaxation of conditioning results in small loss in terms of the rationale for conditional inference, but can greatly reduce the difficulties caused by discreteness. Exact calculation of p-values based on approximate conditioning is possible, but unattractive both in terms of the amount of calculation involved and in requiring explicit specification of the extent to which conditioning is to be relaxed. It is shown that there is a highly accurate, easily computed and very natural asymptotic approximation that avoids these difficulties.

Although the U.S. Food and Drug Administration (FDA, 1992) recommends testing bioequivalence of individual pharmacokinetic parameters one at a time, it seems reasonable and interesting to conduct a test simultaneously for all the parameters. In this paper, we discuss several ways to construct such tests. It is shown that the confidence set approach leads to a test which can be uniformly improved by the intersection of Schuirmann's two one-sided tests procedure. The latter test can further be improved upon noticeably by using the one-dimensional unbiased test of Brown, Hwang & Munk (1997). Numerical calculations of powers are given to support this claim.

SUMMARY Conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed, and have been shown to be equivalent to conventional Bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses. These results are herein extended to sequential testing, and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests. Among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule, seeming to lend frequentist support to the stopping rule principle.

SUMMARY This paper considers a general class of statistics for testing the equality of two survival distributions in clinical trials with sequential monitoring. The tests can be expressed as Lebesgue-Stieltjes integrals of a weight function with respect to the difference between two survival distributions. Prominent members of this class include the two-sample difference in Kaplan-Meier estimates, the test of medians (Brookmeyer & Crowley, 1982), a truncated version of Efron's (1967) test and the Pepe-Fleming statistic (Pepe & Fleming, 1989, 1991). Statistics in this class are shown to converge to a Gaussian process, indexed by information time, under both null and local alternatives even if different statistics are used at different information times. Properly standardised, statistics in a subclass converge to Gaussian processes with independent increments so that the usual group sequential techniques for monitoring a clinical trial can be applied. The design of a trial comparing two treatments with respect to mother-to-newborn transmission of HIV is used to illustrate practical aspects of monitoring.

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.

In this paper we derive Schwarz's information criterion and two modifications for choosing fixed effects in normal linear mixed models. The first modification allows an arbitrary, possibly informative, prior for the parameter of interest. Replacing this prior with the normal, unit-information, prior of Kass & Wasserman (1995) and the generalised Cauchy prior of Jeffreys (1961) yields the usual Schwarz criterion and a second modification, respectively. Under the null hypothesis, these criteria approximate Bayes factors using the corresponding priors to increased accuracy. In regression, the second modification also corresponds asymptotically to the Bayes factors of Zellner & Siow (1980) and O'Hagan (1995), and is similar to the Bayes factor of Berger & Pericchi (1996). In mixed models, the effective sample size term in Schwarz's formula is ambiguous because of correlation between observations. We propose an appropriate generalisation of Schwarz's approximation and apply our results to evaluate a large class of models for repeated neuron area measurements in alcoholic and suicidal patients.

SUMMARY The lack of statistical theory for the planning of early detection trials has resulted in current trials being sub-optimal. We develop probability models that address three characteristics of early detection trials: (i) the optimal time of analysis and length of follow-up, (ii) the optimal spacing between examinations, and (iii) the planning of trials where the numbers of examinations versus sample size are balanced for fixed costs. The optimisation criterion is to maximise the power of the statistical test for comparing mortality. Application is made to breast cancer early detection trials.

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 Bayes/frequentist correspondences between the p-value and the posterior probability of the null hypothesis have been studied in univariate hypothesis testing situations. This paper extends these comparisons to multiple testing and in particular to the Bonferroni multiple testing method, in which p-values are adjusted by multiplying by k, the number of tests considered. In the Bayesian setting, prior assessments may need to be adjusted to account for multiple hypotheses, resulting in corresponding adjustments to the posterior probabilities. Conditions are given for which the adjusted posterior probabilities roughly correspond to Bonferroni adjusted p-values.

We observe a sequence of test statistics T = (T 1 , T 2 , ..., T J ), each of which is approximately N(0,1) under the null hypothesis H 0 , but which are correlated with each other. Not being certain which T j is best, we use the test statistic T max = max{T 1 , ..., T J ). For a given observed value of T max , say c, what is the significance probability pr(T max > c)? Define the length of T to be Σ j L =2 arccos(p j-1,j ), where p j-1,j is the null hypothesis correlation between T j-1 and T j . Hotelling's theorem on the volume of tubes leads to a length-based bound on pr(T max > c) that usually beats the Bonferroni bound. It is easy to improve Hotelling's bound. Several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr(T max > c).

A cutting plane algorithm is proposed for estimating variance and covariance components by maximum likelihood or restricted maximum likelihood enforcing the constraints that covariance matrices be positive semidefinite. For tests of hypotheses involving these constrained estimates, an asymptotic parametric bootstrap is proposed for approximating the distribution of the likelihood ratio test statistic. Although the bootstrap is generally inconsistent when the true parameter value is on the boundary of the feasible region, the double bootstrap can be used to show that the ordinary bootstrap works well in certain problems.

SUMMARY When longitudinal clinical trials are monitored, multiplicity from repeated significance testing as well as from repeated measures has to be accounted for properly to control the overall type I error. This often involves a multidimensional integration procedure to compute group sequential boundaries. We establish an independent increments structure of sequentially computed test statistics based on the generalised estimating equations of Liang & Zeger (1986) for longitudinal data. This simplifies the computational procedure for group sequential boundaries to one involving recursive one-dimensional integrations and allows the use of standard methodology for group sequential tests. We also apply the error spending function approach of Lan & DeMets (1983) by defining information fraction in this setting.

SUMMARY A distribution-free permutation test procedure is proposed for comparing receiver operating characteristic curves based on continuous data from a paired design. The method tests the hypothesis that the two curves are identical for all operating points, unlike previously proposed methods which test the equivalence of the areas under the curves. The new test is shown by simulation to have very similar operating characteristics to the standard method based on comparisons of the areas when the curves are parallel, but markedly superior power when the curves cross, that is when the curves are different but have similar areas. The prospects of generalising the approach to unpaired experiments and to comparisons of ordinal rating data are discussed.

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 Long-range dependence is often observed in long time series. Correlations decay approximately like Ik I2"-2, with H E (0 5, 1), as the lag k tends to infinity. The long-term features of the data are essentially characterised by the parameter H. Small changes of H have strong implications for the long-term behaviour of the process. In particular, rates of convergence of estimators for the mean, and for many other parameters of interest, differ for different values of H. For some data sets, H appears to change with time. In this paper we consider a simple test of the null hypothesis that H is constant. The test is based on a functional central limit theorem for quadratic forms. Critical values for the test statistic are given. Simulations confirm the validity of the test. A data example illustrates its practical application.