The ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables. This goal is, however, seldom achieved because most established regression models estimate only the conditional mean as a function of the explanatory variables and assume that higher moments are not affected by the regressors. The underlying reason for such a restriction is the assumption of additivity of signal and noise. We propose to relax this common assumption in the framework of transformation models. The novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables. These transformation functions are estimated by regularized optimization of scoring rules for probabilistic forecasts, e.g. the continuous ranked probability score. The corresponding estimated conditional distribution functions are consistent. Conditional transformation models are potentially useful for describing possible heteroscedasticity, comparing spatially varying distributions, identifying extreme events, deriving prediction intervals and selecting variables beyond mean regression effects. An empirical investigation based on a heteroscedastic varying-coefficient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more beneficial than kernel-based non-parametric approaches or parametric generalized additive models for location, scale and shape.

Many time series are not second order stationary and it is not appropriate to analyse them by using methods designed for stationary series. The paper introduces a new test for second-order stationarity that detects kinds of departures from stationarity that are different from those based on Fourier methods. The new test is also computationally fast, designed to work with Gaussian and a wide range of non-Gaussian time series, and can locate non-stationarities in time and scale. The test is demonstrated on earthquake, explosion, infant electrocardiogram and simulated time series showing varying degrees of stationarity. The second main contribution develops approximate confidence intervals for time varying autocovariances for locally stationary series as the usual bands computed for stationary series are not appropriate. Our new bands enable practitioners to assess time varying autocovariances statistically and are exhibited on localized autocovariances of explosion and simulated time series.

There are two popular smoothing parameter selection methods for spline smoothing. First, smoothing parameters can be estimated by minimizing criteria that approximate the average mean-squared error of the regression function estimator. Second, the maximum likelihood paradigm can be employed, under the assumption that the regression function is a realization of some stochastic process. The asymptotic properties of both smoothing parameter estimators for penalized splines are studied and compared. A simulation study and a real data example illustrate the theoretical findings.

We describe a model for the analysis of data distributed over irregularly shaped spatial domains with complex boundaries, strong concavities and interior holes. Adopting an approach that is typical of functional data analysis, we propose a spatial spline regression model that is computationally efficient, allows for spatially distributed covariate information and can impose various conditions over the boundaries of the domain. Accurate surface estimation is achieved by the use of piecewise linear and quadratic finite elements.

Interdependent effects are usually distinguished from statistical interaction by using the sufficient causes framework. This almost always involves expressing probability distributions as deterministic logic functions, where certain conditions invariably produce or prevent an outcome. Using an idea from the philosophy literature, that a cause is defined as an event which increases the probability of an outcome, a probabilistic sufficient causes framework is developed here. It expresses distributions with probabilistic logic functions and is used to define interdependence without determinism. The connections, between probabilistic logic and the inequalities which define convex polytopes in the space of the distribution parameters, are given. Interdependence is defined for the response behaviour of individuals, defined by latent variables, and is not directly observable. It is shown that the formulation of the models as geometric objects enables the use of algebraic tools to compute observable constraints for detecting interdependence. Novel constraints are derived for detecting causal interdependence patterns and the corresponding statistical tests are described.

The assumption of log-concavity is a flexible and appealing non-parametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator of a probability mass function. We show that the maximum likelihood estimator is strongly consistent and we derive its pointwise asymptotic theory under both the well-specified and misspecified settings. Our asymptotic results are used to calculate confidence intervals for the true log-concave probability mass function. Both the maximum likelihood estimator and the associated confidence intervals may be easily computed by using the R package logcondiscr. We illustrate our theoretical results by using recent data from the H1N1 pandemic in Ontario, Canada.

Models of dynamic networks—networks that evolve over time—have manifold applications. We develop a discrete time generative model for social network evolution that inherits the richness and flexibility of the class of exponential family random-graph models. The model—a separable temporal exponential family random-graph model—facilitates separable modelling of the tie duration distributions and the structural dynamics of tie formation. We develop likelihood-based inference for the model and provide computational algorithms for maximum likelihood estimation. We illustrate the interpretability of the model in analysing a longitudinal network of friendship ties within a school.

Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several families of max-stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and rescaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko and co-workers. This paper deals with statistical inference for max-stable space–time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters. For regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to ∞. Furthermore, we discuss extensions to irregularly spaced locations. A simulation study shows that the method proposed works well for these models.

Recently proposed outlier robust small area estimators can be substantially biased when outliers are drawn from a distribution that has a different mean from that of the rest of the survey data. This naturally leads one to consider an outlier robust bias correction for these estimators. We develop this idea, proposing two different analytical mean-squared error estimators for the ensuing bias-corrected outlier robust estimators. Simulations based on realistic outlier-contaminated data show that the bias correction proposed often leads to more efficient estimators. Furthermore, the mean-squared error estimation methods proposed appear to perform well with a variety of outlier robust small area estimators.

Network data often take the form of repeated interactions between senders and receivers tabulated over time. A primary question to ask of such data is which traits and behaviours are predictive of interaction. To answer this question, a model is introduced for treating directed interactions as a multivariate point process: a Cox multiplicative intensity model using covariates that depend on the history of the process. Consistency and asymptotic normality are proved for the resulting partial-likelihood-based estimators under suitable regularity conditions, and an efficient fitting procedure is described. Multicast interactions—those involving a single sender but multiple receivers—are treated explicitly. The resulting inferential framework is then employed to model message sending behaviour in a corporate e-mail network. The analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection.

Estimation of high dimensional covariance matrices is known to be a difficult problem, has many applications and is of current interest to the larger statistics community. In many applications including the so-called ‘large p, small n’ setting, the estimate of the covariance matrix is required to be not only invertible but also well conditioned. Although many regularization schemes attempt to do this, none of them address the ill conditioning problem directly. We propose a maximum likelihood approach, with the direct goal of obtaining a well-conditioned estimator. No sparsity assumptions on either the covariance matrix or its inverse are imposed, thus making our procedure more widely applicable. We demonstrate that the proposed regularization scheme is computationally efficient, yields a type of Steinian shrinkage estimator and has a natural Bayesian interpretation. We investigate the theoretical properties of the regularized covariance estimator comprehensively, including its regularization path, and proceed to develop an approach that adaptively determines the level of regularization that is required. Finally, we demonstrate the performance of the regularized estimator in decision theoretic comparisons and in the financial portfolio optimization setting. The approach proposed has desirable properties and can serve as a competitive procedure, especially when the sample size is small and when a well-conditioned estimator is required.

A methodology for the simultaneous Bayesian non-parametric modelling of several distributions is developed. Our approach uses normalized random measures with independent increments and builds dependence through the superposition of shared processes. The properties of the prior are described and the modelling possibilities of this framework are explored in detail. Efficient slice sampling methods are developed for inference. Various posterior summaries are introduced which allow better understanding of the differences between distributions. The methods are illustrated on simulated data and examples from survival analysis and stochastic frontier analysis.

Determining how to select the tuning parameter appropriately is essential in penalized likelihood methods for high dimensional data analysis. We examine this problem in the setting of penalized likelihood methods for generalized linear models, where the dimensionality of covariates p is allowed to increase exponentially with the sample size n. We propose to select the tuning parameter by optimizing the generalized information criterion with an appropriate model complexity penalty. To ensure that we consistently identify the true model, a range for the model complexity penalty is identified in the generlized information criterion. We find that this model complexity penalty should diverge at the rate of some power of  log (p) depending on the tail probability behaviour of the response variables. This reveals that using the Akaike information criterion or Bayes information criterion to select the tuning parameter may not be adequate for consistently identifying the true model. On the basis of our theoretical study, we propose a uniform choice of the model complexity penalty and show that the approach proposed consistently identifies the true model among candidate models with asymptotic probability 1. We justify the performance of the procedure proposed by numerical simulations and a gene expression data analysis.

To date, only frequentist, Bayesian and empirical Bayes approaches have been studied for the large-scale inference problem of testing simultaneously hundreds or thousands of hypotheses. Their derivations start with some summarizing statistics without modelling the basic responses. As a consequence testing procedures have been developed without necessarily checking model assumptions, and empirical null distributions are needed to avoid the problem of rejecting all null hypotheses when the sample sizes are large. Nevertheless these procedures may not be statistically efficient. We present the multiple-testing problem as a multiple-prediction problem of whether a null hypothesis is true or not. We introduce hierarchical random-effect models for basic responses and show how the extended likelihood is built. It is shown that the likelihood prediction has a certain oracle property. The extended likelihood leads to new testing procedures, which are optimal for the usual loss function in hypothesis testing. The new tests are based on certain shrinkage t-statistics and control the local probability of false discovery for individual tests to maintain the global frequentist false discovery rate and have no need to consider an empirical null distribution for the shrinkage t-statistics. Conditions are given when these false rates vanish. Three examples illustrate how to use the likelihood method in practice. A numerical study shows that the likelihood approach can greatly improve existing methods and finding the best fitting model is crucial for the behaviour of test procedures.

We propose a fast penalized spline method for bivariate smoothing. Univariate P-spline smoothers are applied simultaneously along both co-ordinates. The new smoother has a sandwich form which suggested the name ‘sandwich smoother’ to a referee. The sandwich smoother has a tensor product structure that simplifies an asymptotic analysis and it can be fast computed. We derive a local central limit theorem for the sandwich smoother, with simple expressions for the asymptotic bias and variance, by showing that the sandwich smoother is asymptotically equivalent to a bivariate kernel regression estimator with a product kernel. As far as we are aware, this is the first central limit theorem for a bivariate spline estimator of any type. Our simulation study shows that the sandwich smoother is orders of magnitude faster to compute than other bivariate spline smoothers, even when the latter are computed by using a fast generalized linear array model algorithm, and comparable with them in terms of mean integrated squared errors. We extend the sandwich smoother to array data of higher dimensions, where a generalized linear array model algorithm improves the computational speed of the sandwich smoother. One important application of the sandwich smoother is to estimate covariance functions in functional data analysis. In this application, our numerical results show that the sandwich smoother is orders of magnitude faster than local linear regression. The speed of the sandwich formula is important because functional data sets are becoming quite large.