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Highlights, trends and influences are identified associated with the pages of Biometrika subsequent to the editorship of Karl Pearson. [ABSTRACT FROM PUBLISHER]

Interpoint distances have applications in many areas of probability and statistics. Thanks to their simplicity of computation, interpoint distance-based procedures are particularly appealing for analysing small samples of high-dimensional data. In this paper, we first study the asymptotic distribution of interpoint distances in the high-dimension, low-sample-size setting and show that it is normal under regularity conditions. We then construct a powerful test for the two-sample problem, which is consistent for detecting location and scale differences. Simulations show that the test compares favourably with existing distance-based tests. [ABSTRACT FROM AUTHOR]

We consider the problem of using data in studies with an unobserved treatment/response variable in order to evaluate average causal effects, when cause-effect relationships between variables can be described by a directed acyclic graph and the corresponding recursive factorization of a joint distribution. The paper proposes graphical criteria to test whether average causal effects are identifiable even if a treatment/response variable is unobserved. If the answer is affirmative, we provide further formulations for average causal effects from the observed data. The graphical criteria enable us to evaluate average causal effects when it is difficult to observe a treatment/response variable. [ABSTRACT FROM AUTHOR]

Sampling related to the outcome variable of a regression analysis conditional on covariates is called informative sampling and may lead to bias in ordinary least squares estimation. Weighting by the reciprocal of the inclusion probability approximately removes such bias but may inflate variance. This paper investigates two ways of modifying such weights to improve efficiency while retaining consistency. One approach is to multiply the inverse probability weights by functions of the covariates. The second is to smooth the weights given values of the outcome variable and covariates. Optimal ways of constructing weights by these two approaches are explored. Both approaches require the fitting of auxiliary weight models. The asymptotic properties of the resulting estimators are investigated and linearization variance estimators are obtained. The approach is extended to pseudo maximum likelihood estimation for generalized linear models. The properties of the different weighted estimators are compared in a limited simulation study. The robustness of the estimators to misspecification of the auxiliary weight model or of the regression model of interest is discussed. [ABSTRACT FROM PUBLISHER]

This paper considers the construction of blocked two-level regular designs with weak minimum aberration. We first obtain the minimum value of the number of two-factor interactions which are aliased with the block effects. Based on this result, two methods are then proposed in two different scenarios to construct weak minimum aberration blocked two-level designs with respect to some existing combined wordlength patterns. [ABSTRACT FROM PUBLISHER]

We present a general class of nonlinear regression and time series models that we call generalised structured models. The class is a natural generalisation of generalised additive models, and it includes generalised interaction models, structured volatility models, visual GARCH, generalised autoregressive conditional heteroscedasticity, models and varying coefficient models. We discuss estimation principles including smoothing splines and a generalisation of the projection approach of Mammen et al. (1999). We finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new stochastic variant of the EM algorithm. The algorithm combines the principle of multiple imputation and the theory of simulated annealing to deal with cases where the E-step and the M-step can be intractable or numerically inefficient. [ABSTRACT FROM AUTHOR]

For a statistical model with data, likelihood for the scalar or vector full parameter θ, of dimension p say, is typically well defined and easily computed. In this paper, we investigate likelihood for a component parameter ψ(θ) of dimension d < p and make use of the recent likelihood theory that has been successful in producing highly accurate third-order p-values for scalar parameters of continuous models. The theory leads under moderate regularity to a definitive third-order determination of likelihood for a component parameter ψ(θ) of dimension d, where 1 ≤ d ≤ p. We use the simple location model on the plane with standard normal errors to motivate the development. The example exhibits most of the key characteristics of the general case and the recent theory then extends the determination of likelihood to the general context. For the scalar interest parameter case with d = 1, the usual determinations are typically of second-order accuracy; the example indicates how the new determination achieves third-order accuracy. The implementation is straightforward and uses familiar ingredients to other determinations, such as the full maximum likelihood value θ , the constrained value θ ~ψ given ψ(θ) = ψ, and the observed information jλλ(θ ψ) for a complementing nuisance parameter λ(θ). It does however require a special version of the nuisance information jλλ(θ ψ), a version calibrated relative to a symmetric choice of the exponential-type reparameterisation φ(θ) underlying the recent theory, but this is easily computed. Various examples are given and the motivating example is discussed in detail. [ABSTRACT FROM AUTHOR]

There has been much research on analysing multivariate failure times, but little that has accommodated failures that arise in the presence of a competing failure process. This paper studies the problem of describing associations among times to such failures. It proposes a modified conditional hazard ratio measure of association that is tailored to competing risks data, develops frailty models and a nonparametric method for describing the proposed measure, and contrasts estimation by proposed methods with the ‘standard’ of treating competing risks as independently censoring failure times due to targeted causes. The methods are investigated on simulated and real data. [ABSTRACT FROM PUBLISHER]

In a parametric model the maximum likelihood estimator of a parameter of interest ψ may be viewed as the solution to the equation l′p(ψ) = 0, where lp denotes the profile loglikelihood function. It is well known that the estimating function l′p(ψ) is not unbiased and that this bias can, in some cases, lead to poor estimates of ψ. An alternative approach is to use the modified profile likelihood function, or an approximation to the modified profile likelihood function, which yields an estimating function that is approximately unbiased. In many cases, the maximum likelihood estimating functions are unbiased under more general assumptions than those used to construct the likelihood function, for example under first‐ or second‐moment conditions. Although the likelihood function itself may provide valid estimates under moment conditions alone, the modified profile likelihood requires a full parametric model. In this paper, modifications to l′p(ψ) are presented that yield an approximately unbiased estimating function under more general conditions. [ABSTRACT FROM PUBLISHER]

Adaptive designs are often used in clinical trials to force balance in sequential allocation of patients between two or more competitive treatments. Sometimes, from ethical considerations, the goal may be to allocate a larger number of patients to the better treatment in the course of the trial. In the present paper we consider patients who are heterogeneous with respect to some prognostic factors and assume that the response from each patient is a continuous variable. We consider a normal linear model and provide an allocation design with due attention to the prognostic factors. The loss of efficiency incurred by using a balanced design is also indicated. Issues of inference based on adaptive allocation are also discussed. Finally a guideline is provided for the appropriate choice of the design parameters. [ABSTRACT FROM PUBLISHER]

We consider parametric frameworks for the prediction of future values of a random variable Y, based on previously observed data X. Simple pivotal methods for obtaining calibrated prediction intervals are presented and illustrated. Frequentist predictive distributions are defined as confidence distributions, and their utility is demonstrated. A simple pivotal-based approach that produces prediction intervals and predictive distributions with well-calibrated frequentist probability interpretations is introduced, and efficient simulation methods for producing predictive distributions are considered. Properties related to an average Kullback–Leibler measure of goodness for predictive or estimated distributions are given. The predictive distributions here are shown to be optimal in certain settings with invariance structure, and to dominate plug-in distributions under certain conditions. [ABSTRACT FROM AUTHOR]

Replicability analysis aims to identify the overlapping signals across independent studies that examine the same features. For this purpose we develop hypothesis testing procedures that first select the promising features from each of two studies separately. Only those features selected in both studies are then tested. The proposed procedures have theoretical guarantees regarding their control of the familywise error rate or false discovery rate on the replicability claims. They can also be used for signal discovery in each study separately, with the desired error control. Their power for detecting truly replicable findings is compared to alternatives. We illustrate the procedures on behavioural genetics data. [ABSTRACT FROM AUTHOR]

We derive a structural break test for extremal dependence in $$\beta$$ -mixing, possibly high-dimensional random vectors with either asymptotically dependent or asymptotically independent components. Existing tests require serially independent observations with asymptotically dependent components. To avoid estimating a long-run variance, we use self-normalization, which obviates the need to estimate the coefficient of tail dependence when components are asymptotically independent. Simulations show favourable empirical size and power of the test, which we apply to S&P 500 and DAX log-returns. We find evidence for one break in the coefficient of tail dependence for the upper and lower joint tail at the beginning of the 2007–08 financial crisis, leading to more extremal co-movement. [ABSTRACT FROM AUTHOR]

The seminal work of Morgan & Rubin (2012) considers rerandomization for all the units at one time.In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result that given the same number of rerandomizations, in expected value, under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time. [ABSTRACT FROM AUTHOR]

Linearity, sometimes jointly with constant variance, is routinely assumed in the context of sufficient dimension reduction. It is well understood that, when these conditions do not hold, blindly using them may lead to inconsistency in estimating the central subspace and the central mean subspace. Surprisingly, we discover that even if these conditions do hold, using them will bring efficiency loss. This paradoxical phenomenon is illustrated through sliced inverse regression and principal Hessian directions. The efficiency loss also applies to other dimension reduction procedures. We explain this empirical discovery by theoretical investigation. [ABSTRACT FROM PUBLISHER]

We consider forecasting from age-period-cohort models, as well as from the extended chain-ladder model. The parameters of these models are known only to be identified up to linear trends. Forecasts from such models may therefore depend on arbitrary linear trends. A condition for invariant forecasts is proposed. A number of standard forecast models are analysed. [ABSTRACT FROM AUTHOR]

We present a method for multiple hypothesis testing that maintains control of the false discovery rate while incorporating prior information about the hypotheses. The prior information takes the form of p-value weights. If the assignment of weights is positively associated with the null hypotheses being false, the procedure improves power, except in cases where power is already near one. Even if the assignment of weights is poor, power is only reduced slightly, as long as the weights are not too large. We also provide a similar method for controlling false discovery exceedance. [ABSTRACT FROM AUTHOR]

The linear step-up multiple testing procedure controls the false discovery rate at the desired level q for independent and positively dependent test statistics. When all null hypotheses are true, and the test statistics are independent and continuous, the bound is sharp. When some of the null hypotheses are not true, the procedure is conservative by a factor which is the proportion m0/m of the true null hypotheses among the hypotheses. We provide a new two-stage procedure in which the linear step-up procedure is used in stage one to estimate m0, 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. [ABSTRACT FROM AUTHOR]

In settings where the full probability model is not specified, consider a general estimating function g(θ, λ; y) that involves not only the parameters of interest, θ, but also some nuisance parameters, λ. We consider methods for reducing the effects on g of fitting nuisance parameters. We propose Cox-Reid-type adjustment to the profile estimating function, g(θ, λ ˆθ; y), that reduces its bias by two orders. Typically, only the first two moments of the response variable are needed to form the adjustment. Important applications of this method include the estimation of the pairwise association and main effects in stratified, clustered data and estimation of the main effects in a matched pair study. A brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters. [ABSTRACT FROM AUTHOR]

It is well known, at least through many examples, that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood. Ordinary asymptotics almost totally fail to deal with this issue. For this reason, we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two-index asymptotics setting. This means that both the sample size of the strata, m, and the dimension of the nuisance parameter, q, may increase to infinity. It is shown that in this asymptotic setting modified profile likelihoods give improvements, with respect to the profile likelihood, in terms of consistency of estimators and of asymptotic distributional properties. In particular, the modified profile likelihood based statistics have the usual asymptotic distribution, provided that 1/m = o(q-1/3), while the analogous condition for the profile likelihood is 1/m = o(q-1). [ABSTRACT FROM AUTHOR]

The asymptotic variance and robust variance estimators of rate ratios estimated using conditional logistic regression from individually-matched case-control data are derived when the presumed proportional hazards model is misspecified. The robust variance estimators are easily computed using Schoenfeld residuals generated from standard partial likelihood estimation software for failure time data. Simulation studies indicate that the robust variance estimators perform well for typical sizes and that the 'rare disease' version should be adequate for all practical purposes. It was also found that model misspecification must be quite extreme before the model-based, i.e. inverse information, variance is significantly biased and that the robust variance estimators are somewhat more variable than the model-based. We conclude that the model-based variance estimator can be used when model misspecification is not severe. The robust estimator should be used when the presumed model clearly fits the data poorly. [ABSTRACT FROM AUTHOR]

Edwards & Lauritzen (2001) have recently proposed the TM algorithm for finding the maximum likelihood estimate when the likelihood can be truly or artificially regarded as a conditional likelihood, and the full likelihood is more easily maximised. They have presented a proof of convergence, provided that the algorithm is supplemented by a line search. In this note a simple expression, in terms of observed information matrices, is given for the convergence rate of the algorithm per se, when it converges, and the result elucidates also in which situations the algorithm will require a line search. Essentially these are cases when the full model does not adequately fit the data. [ABSTRACT FROM PUBLISHER]

It is shown that both the simple form of the Rasch model for binary data and a generalisation are essentially equivalent to special dichotomised Gaussian models. In these the underlying Gaussian structure is of single factor form; that is, the correlations between the binary variables arise via a single underlying variable, called in psychometrics a latent trait. The implications for scoring of the binary variables are discussed, in particular regarding the scoring system as in effect estimating the latent trait. In particular, the role of the simple sum score, in effect the total number of ‘successes’, is examined. Relations with the principal component analysis of binary data are outlined and some connections with the quadratic exponential binary model are sketched. [ABSTRACT FROM PUBLISHER]

We present a new additive‐multiplicative hazard model which consists of two components. The first component contains additive covariate effects through an additive Aalen model while the second component contains multiplicative covariate effects through a Cox regression model. The Aalen model allows for time‐varying covariate effects, while the Cox model allows only a common time‐dependence through the baseline. Approximate maximum likelihood estimators are derived by solving the simultaneous score equations for the nonparametric and parametric components of the model. The suggested estimators are provided with large‐sample properties and are shown to be efficient. The efficient estimators depend, however, on some estimated weights. We therefore also consider unweighted estimators and describe their large‐sample properties. We finally extend the model to allow for time‐varying covariate effects in the multiplicative part of the model as well. [ABSTRACT FROM PUBLISHER]

We review objective Bayes procedures based on both parametric and predictive coverage probability bias and explore the extent to which such procedures contravene the likelihood principle in the case of a scalar parameter. The discussion encompasses choice of objective priors, objective posterior probability statements and objective predictive probability statements. We conclude with some remarks concerning the future development and implementation of objective priors based on small coverage probability bias. [ABSTRACT FROM PUBLISHER]

A necessary and sufficient condition for time-reversibility of stationary linear processes is given which, contrary to previous results, does not require existence of moments of order higher than two. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]