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Olkin & Vaeth (1981) consider a two-way classification model with k rows and p columns in which the residuals (ei1, ..., eip) for row i are independently distributed with a multivariate normal distribution with zero means and common covariance matrix ((ij)). They concentrate their attention particularly on the situation where the row classification corresponds to k subjects, each treated in one of two or more ways; the column classification corresponds to p repetitions, perhaps in time, from each subject, probably resulting in correlated errors. Replication of subjects within treatments enables the elements of the covariance matrix to be estimated from the data, which in turn enables maximum likelihood estimates of the row and column parameters to be computed and likelihood ratio tests to be carried out without having to make assumptions about the form of the covariance matrix. The careful analysis of data of this kind has been the subject of numerous papers, Wishart (1938) perhaps being the first to concentrate on this topic. Despite this, however, the use of erroneous methods in published material is widespread and this provided the stimulus for the publication of our earlier paper (Rowell & Walters, 1976), and also for the present communication. We note here that there is confusion in Olkin & Vaeth's row/column terminology in their worked example: the rows in their tables are referred to as columns in the text, and vice versa. In what follows, when referring to their numerical example, the rows classification will refer to the high/low factor, and the columns classification to the time factor.

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights "minimal weights" and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance., 41 pages

In this paper we consider a location model of the form $Y = m(X) + \varepsilon$, where $m(\cdot)$ is the unknown regression function, the error $\varepsilon$ is independent of the $p$-dimensional covariate $X$ and $E(\varepsilon)=0$. Given i.i.d. data $(X_1,Y_1),\ldots,(X_n,Y_n)$ and given an estimator $\hat m(\cdot)$ of the function $m(\cdot)$ (which can be parametric or nonparametric of nature), we estimate the distribution of the error term $\varepsilon$ by the empirical distribution of the residuals $Y_i-\hat m(X_i)$, $i=1,\ldots,n$. To approximate the distribution of this estimator, Koul and Lahiri (1994) and Neumeyer (2008, 2009) proposed bootstrap procedures, based on smoothing the residuals either before or after drawing bootstrap samples. So far it has been an open question whether a classical non-smooth residual bootstrap is asymptotically valid in this context. In this paper we solve this open problem, and show that the non-smooth residual bootstrap is consistent. We illustrate this theoretical result by means of simulations, that show the accuracy of this bootstrap procedure for various models, testing procedures and sample sizes.

The envelope model allows efficient estimation in multivariate linear regression. In this paper, we propose the sparse envelope model, which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients. The envelope estimator is consistent but not sparse, and in many situations it is important to identify the response variables for which the regression coefficients are zero. The sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model. Response variable selection arises naturally in many applications, but has not been studied as thoroughly as predictor variable selection. In this paper, we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts. In response variable selection, even if a response has zero coefficients, it should still be retained to improve the estimation efficiency of the nonzero coefficients. This is different from the practice in predictor variable selection. We establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator.

In a recent paper Dette et al. (2014) introduced optimal design problems for dose finding studies with an active control. These authors concentrated on regression models with normal distributed errors (with known variance) and the problem of determining optimal designs for estimating the smallest dose, which achieves the same treatment effect as the active control. This paper discusses the problem of designing active-controlled dose finding studies from a broader perspective. In particular, we consider a general class of optimality criteria and models arising from an exponential family, which are frequently used analyzing count data. We investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control. Optimal designs are constructed for several situations and the differences arising from different distributional assumptions are investigated in detail. In particular, our results are applicable for constructing optimal experimental designs to analyze active-controlled dose finding studies with discrete data, and we illustrate the efficiency of the new optimal designs with two recent examples from our consulting projects.

This paper proposes a unified framework to characterize the rate function of a recurrent event process through shape and size parameters. In contrast to the intensity function, which is the event occurrence rate conditional on the event history, the rate function is the occurrence rate unconditional on the event history, and thus it can be interpreted as a population-averaged count of events in unit time. In this paper, shape and size parameters are introduced and used to characterize the association between the rate function λ(⋅) and a random variable X. Measures of association between X and λ(⋅) are defined via shape- and size-based coefficients. Rate-independence of X and λ(⋅) is studied through tests of shape-independence and size-independence, where the shape- and size-based test statistics can be used separately or in combination. These tests can be applied when X is a covariable possibly correlated with the recurrent event process through λ(⋅) or, in the one-sample setting, when X is the censoring time at which the observation of N(⋅) is terminated. The proposed tests are shape- and size-based, so when a null hypothesis is rejected, the test results can serve to distinguish the source of violation.

This paper deals with the design of the likelihood continual reassessment method, which is an increasingly widely used model-based method for dose-finding studies. It is common to implement the method in a two-stage approach, whereby the model-based stage is activated after an initial sequence of patients has been treated. While this two-stage approach is practically appealing, it lacks a theoretical framework, and it is often unclear how the design components should be specified. This paper develops a general framework based on the coherence principle, from which we derive a design calibration process. A real clinical-trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner, while offering competitive operating characteristics. We explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose-toxicity model.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics.

This paper considers benchmarking issues in the context of small area estimation. We find optimal estimators within the class of benchmarked linear estimators under linear constraints. This extends existing results for external and internal benchmarking, and also links the two. Necessary and sufficient conditions for self-benchmarking are found for an augmented model. Most results of this paper are found using ideas of orthogonal projection Copyright 2013, Oxford University Press.

In some problems involving functional data, it is desired to undertake prediction or classification before the full trajectory of a function is observed. In such cases, it is often preferable to suffer somewhat greater error in return for making a decision relatively early. The prediction and classification problems can be treated similarly, using mean squared prediction error, or classification error, respectively, as the means for quantifying performance, so in this paper we focus principally on classification. We introduce a method for determining when an early decision can reasonably be made, using only part of the trajectory, and we show how to use the method to choose among data types. Our approach is fully nonparametric, and no specific model is required. Properties of error-rate are studied as functions of time and data type. The effectiveness of the proposed method is illustrated in both theoretical and numerical terms. The classification referred to in this paper would be termed supervised classification in machine learning, to distinguish it from unsupervised classification, or clustering. Copyright 2012, Oxford University Press.

This paper derives the asymptotic null distribution of a quasilikelihood ratio test statistic for an autoregressive moving average model against its threshold extension. The null hypothesis is that of no threshold, and the error term could be dependent. The asymptotic distribution is rather complicated, and all existing methods for approximating a distribution in the related literature fail to work. Hence, a novel bootstrap approximation based on stochastic permutation is proposed in this paper. Besides being robust to the assumptions on the error term, our method enjoys more flexibility and needs less computation when compared with methods currently used in the literature. Monte Carlo experiments give further support to the new approach, and an illustration is reported. Copyright 2011, Oxford University Press.

In recent years, several authors have investigated response-adaptive allocation rules for comparative clinical trials, in order to favour, at each stage of the trial, the treatment that appears to be best. In this paper, we define admissible allocations, namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion, reflecting the inferential properties of the experiment, and the proportion of patients assigned to the best treatment or treatments; we survey existing designs from this viewpoint. We also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria, with weights that depend on the actual treatment effects. This compound criterion leads to a locally optimal allocation that can be targeted by some response-adaptive randomization rule. The paper mainly deals with the case of two treatments, but the suggested methodology is shown to extend to more than two. Copyright 2010, Oxford University Press.

In this paper we discuss the analysis of multi-phase, or multi-stage, case-control studies and present an efficient semiparametric maximum-likelihood approach that unifies and extends earlier work, including the seminal case-control paper by Prentice & Pyke (1979), work by Breslow & Cain (1988), Scott & Wild (1991), Breslow & Holubkov (1997) and others. The theoretical derivations apply to arbitrary binary regression models but we present results for logistic regression and show that the approach can be implemented by including additional intercept terms in the logistic model and then making some simple corrections to the score and information equations used in a Newton--Raphson or Fisher-scoring maximization of the prospective loglikelihood. Copyright 2010, Oxford University Press.

A Gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non-Gaussian. Our paper presents a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous marginals, and generalises Gaussian graphical models to the Gaussian copula framework. Posterior inference is carried out using a novel and efficient simulation method. The methods in the paper are applied to simulated and real data.

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 Properties of the periodogram are seldom studied in the setting of nonparametric regression, although that is the context in which the periodogram is widely applied in astronomy. There it is a competitor with more recent least-squares methods. The periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem. However, as we show in the present paper, it also has drawbacks. The estimator that it produces has somewhat higher variance than its least-squares counterpart, and a periodogram-based approach is more prone to suffer difficulties caused by periodicity of the observation schedule. While the periodogram remains a very attractive tool, the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting. This aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset.

Our paper proposes adaptive Monte Carlo sampling schemes for Bayesian variable selection in linear regression that improve on standard Markov chain methods. We do so by considering Metropolis-Hastings proposals that make use of accumulated information about the posterior distribution obtained during sampling. Adaptation needs to be done carefully to ensure that sampling is from the correct ergodic distribution. We give conditions for the validity of an adaptive sampling scheme in this problem, and for simulating from a distribution on a finite state space in general, and suggest a class of adaptive proposal densities which uses best linear prediction to approximate the Gibbs sampler. Our sampling scheme is computationally much faster per iteration than the Gibbs sampler, and when this is taken into account the efficiency gains when using our sampling scheme compared to alternative approaches are substantial in terms of precision of estimation of posterior quantities of interest for a given amount of computation time. We compare our method with other sampling schemes for examples involving both real and simulated data. The methodology developed in the paper can be extended to variable selection in more general problems.

The recent paper by Peng & Yao (2003) gave an interesting extension of least absolute deviation estimation to generalised autoregressive conditional heteroscedasticity, GARCH, time series models. The asymptotic distributions of absolute residual autocorrelations and squared residual autocorrelations from the GARCH model estimated by the least absolute deviation method are derived in this paper. These results lead to two useful diagnostic tools which can be used to check whether or not a GARCH model fitted by using the least absolute deviation method is adequate. Some simulation experiments give further support to the asymptotic theory and a real data example is also reported.

SUMMARY This paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters. In a statistical model with unknown nuisance parameters, the efficiency of an estimator of a parameter usually increases when the nuisance para meters are known. However the opposite phenomenon can sometimes occur. In this paper, we elucidate the occurrence of this paradox by examining estimating functions. In particular, we focus on the projected estimating function, which is defined by the projection of the score function on to a given estimating function. A sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters. In addition, a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators.

SUMMARY In this paper we consider optimal design of experiments in the case of correlated obser- vations. We use and further develop the concept of design measures introduced by Paizman & Mtiller (1998) for the construction of a simple, quick and elegant design algorithm. We support the construction of this algorithm for a general correlation structure by an interpretation in terms of norms. Examples demonstrate that our results are useful for generating exact designs by sampling from the obtained design measures. Most of the literature on experimental design operates under the assumption of uncorre- lated errors and by exploiting the concept of a design measure introduced by Kiefer (1959). In this setting an optimal design measure is usually computed by an iterative algorithm and then an exact design with independent replications approximately proportional to that measure is employed. In the correlated case when replications are not allowed the implementation of design measures is not so straightforward. Recently, however, by adding virtual design-dependent noise to the process, Paizman & Muiller (1998) have introduced a way of using this popular concept, albeit requiring a very different interpretation. Furthermore, Paizman & Muller (2000) showed that solving minimisation problems in terms of these measures corresponds to computation of exact designs. These measures serve as building blocks for the methods presented in this paper, which provides a general numerical tool for screening designs. By a different type of smoothing of some important nondifferentiable terms in the algorithm, and without imposing restric- tions on the number of support points, we obtain a design measure. The relative magnitude of this measure for each support point reflects the importance of its inclusion in any exact design. This algorithm can also be interpreted in terms of certain 'information' norms as pre

SUMMARY This paper studies simultaneous feature selection and extraction in supervised and unsupervised learning. We propose and investigate selective reduced rank regression for constructing optimal explanatory factors from a parsimonious subset of input features. The proposed estimators enjoy sharp oracle inequalities, and with a predictive information criterion for model selection, they adapt to unknown sparsity by controlling both rank and row support of the coefficient matrix. A class of algorithms is developed that can accommodate various convex and nonconvex sparsity-inducing penalties, and can be used for rank-constrained variable screening in high-dimensional multivariate data. The paper also showcases applications in macroeconomics and computer vision to demonstrate how low-dimensional data structures can be effectively captured by joint variable selection and projection.

Summary In this article we develop a method based on model-X knockoffs to find conditional associations that are consistent across environments, while controlling the false discovery rate. The motivation for this problem is that large datasets may contain numerous associations that are statistically significant and yet misleading, as they are induced by confounders or sampling imperfections. However, associations replicated under different conditions may be more interesting. In fact, sometimes consistency provably leads to valid causal inferences even if conditional associations do not. Although the proposed method is widely applicable, in this paper we highlight its relevance to genome-wide association studies, in which robustness across populations with diverse ancestries mitigates confounding due to unmeasured variants. The effectiveness of this approach is demonstrated by simulations and applications to UK Biobank data.

Summary Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much attention recently. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjee’s new correlation coefficient with three established rank correlations that also facilitate consistent tests of independence, namely Hoeffding’s $D$, Blum–Kiefer–Rosenblatt’s $R$, and Bergsma–Dassios–Yanagimoto’s $\tau^*$. We compare the computational efficiency of these rank correlation coefficients in light of recent advances, and investigate their power against local rotation and mixture alternatives. Our main results show that Chatterjee’s coefficient is unfortunately rate-suboptimal compared to $D$, $R$ and $\tau^*$. The situation is more subtle for a related earlier estimator of Dette et al. (2013). These results favour $D$, $R$ and $\tau^*$ over Chatterjee’s new correlation coefficient for the purpose of testing independence.

We consider two-level designs in this paper, with the two levels denoted by + 1 and -1. The run size n of a supersaturated design does not exceed its number of columns, or factors, m, that is n < m. In a screening experiment, although a large number of factors are studied, the number of active factors is usually very small. This assumption of effect sparsity (Box & Meyer, 1986) is the rationale for using supersaturated designs. Booth & Cox (1962) was the first paper to consider deterministic construction of supersaturated designs. Three systematic construction methods were recently proposed and discussed in Lin (1993), Wu (1993) and Tang & Wu (1997), and computer generation of supersaturated designs was considered by Lin (1995), Nguyen (1996) and Li & Wu (1997). Since n < m, supersaturated designs cannot be made orthogonal. It is therefore desirable to have supersaturated designs which are as close to orthogonal as possible. The most commonly used measure of non-orthogonality among the columns of a supersaturated design is

Recent papers by Barndorff-Nielsen & Cox (1996) and Vidoni (1998) concern the determination of prediction limits for an unobserved continuous random variable, for which the coverage probability equals the target value to third-order accuracy. These results are usually expressed by means of the predictive density which generates the required prediction limit as the corresponding quantile. This paper discusses first the case where the variable to be predicted is discrete. This case requires special consideration, since a direct application of the above mentioned results is not possible. Secondly, these prediction limits, and the corresponding predictive distributions, are computed for both discrete and continuous generalised linear models.

This paper addresses the task of locating and identifying an unknown number of objects of different types in an image. Baddeley & Van Lieshout (1993) advocate marked point processes as object priors, whereas Grenander & Miller (1994) use deformable template models. In this paper elements of both approaches are combined to handle scenes containing variable numbers of objects of different types, using reversible jump Markov chain Monte Carlo methods for inference (Green, 1995). The naive application of these methods here leads to slow mixing and we adapt the model and algorithm in tandem in proposing three strategies to deal with this. The first two expand the model space by introducing an additional 'unknown' object type and the idea of a variable resolution template. The third strategy, utilising the first two, augments the algorithm with classes of updates which provide intuitive transitions between realisations containing different numbers of cells by splitting or merging nearby objects.

SUMMARY We put forward a state space model where the unobservable state variable can be any Gaussian stochastic process. We discuss both maximum likelihood estimation and Bayesian inference for this generalised model. The methodology developed in this paper is particularly important for the class of long memory plus noise models. Armed with the simulation smoother introduced in this paper, we can estimate a class of non-Gaussian measurement time series models with long memory in the state equation.

In this paper we discuss maximum likelihood estimation when some observations are missing in mixed graphical interaction models assuming a conditional Gaussian distribution as introduced by Lauritzen&Wermuth (1989). For the saturated case ML estimation with missing values via the EM algorithm has been proposed by Little&Schluchter (1985). We expand their results to the special restrictions in graphical models and indicate a more efficient way to compute the E--step. The main purpose of the paper is to show that for certain missing patterns the computational effort can considerably be reduced.

Summary As a generalization of the classical linear factor model, generalized latent factor models are useful for analysing multivariate data of different types, including binary choices and counts. This paper proposes an information criterion to determine the number of factors in generalized latent factor models. The consistency of the proposed information criterion is established under a high-dimensional setting, where both the sample size and the number of manifest variables grow to infinity, and data may have many missing values. An error bound is established for the parameter estimates, which plays an important role in establishing the consistency of the proposed information criterion. This error bound improves several existing results and may be of independent theoretical interest. We evaluate the proposed method by a simulation study and an application to Eysenck’s personality questionnaire.

Consider inference about a scalar parameter of interest θ in the presence of a vector nuisance parameter. Inference about θ is often based on a pseudolikelihood function. In this paper, the general problem of constructing a pseudo-loglikelihood function H(θ) is considered. Conditions are given under which H has the same properties as a genuine loglikelihood function for a model without a nuisance parameter. When these conditions are satisfied to a given order of approximation, H is said to be a jth-order local loglikelihood function. The theory of local loglikelihood functions is developed and it is shown that second-order versions of these have a number of desirable properties. Several commonly used pseudolikelihood functions are studied from this point of view. One commonly used pseudolikelihood function is profile likelihood in which parameters other than 0 are replaced by their maximum likelihood estimates. A second aspect of the paper is to consider the use of other estimates in this context. Examples are given which suggest that inference about θ may be improved if a method other than maximum likelihood is used, particularly when the number of other parameters is large relative to the sample size.

SUMMARY This paper examines the impact of randomisation and blinding in experiments in which subjects act like investigators by attempting to learn about the effectiveness of the administered treatments. We derive the effects that this type of subject inference has on investigator inference. We show that the conditions under which randomisation and blinding induce unbiased estimation when subjects make treatment inferences are extremely strong and unlikely to hold in most experiments. A test for the presence of such subject sampling in blind experiments is proposed, with empirical results from a set of blind clinical trials indicating the occurrence of subject sampling in about one-third of the trials. In experiments, investigators apply statistical analysis to the outcome data to draw inferences about the effects of treatments. This paper hypothesises that subjects, like investigators, make inferences about treatment efficacy from the outcomes that they experience. For example, in an HIV trial the primary outcome may be the strength of a subject's immune system, as measured by CD4 counts. As the trial progresses, each subject, along with the investigator, learns more about the effectiveness of the assigned treatment each successive period by observing the change in his CD4 count. The likelihood that a subject remains in the trial or complies with the treatment protocol thus increases as the health of the subject improves. We refer to this process of subjects making inferences as subject