Summary We study the problem of testing equivalence of functional parameters, such as the mean or the variance function, in the two-sample functional data setting. In contrast to previous work where the functional problem is reduced to a multiple testing problem for the equivalence of scalar data by comparing the functions at each point, our approach is based on an estimate of a distance measuring the maximum deviation between the two functional parameters. Equivalence is claimed if the estimate for the maximum deviation does not exceed a given threshold. We propose a bootstrap procedure for obtaining quantiles of the distribution of the test statistic, and we prove consistency of the corresponding test in the large-sample scenario. As the methods proposed here avoid the use of the intersection-union principle, they are less conservative and more powerful than currently available approaches.

Summary High-dimensional statistical inference with general estimating equations is challenging and remains little explored. We study two problems in the area: confidence set estimation for multiple components of the model parameters, and model specifications tests. First, we propose to construct a new set of estimating equations such that the impact from estimating the high-dimensional nuisance parameters becomes asymptotically negligible. The new construction enables us to estimate a valid confidence region by empirical likelihood ratio. Second, we propose a test statistic as the maximum of the marginal empirical likelihood ratios to quantify data evidence against the model specification. Our theory establishes the validity of the proposed empirical likelihood approaches, accommodating over-identification and exponentially growing data dimensionality. Numerical studies demonstrate promising performance and potential practical benefits of the new methods.

Summary We study posterior contraction rates in sparse high-dimensional generalized linear models using priors incorporating sparsity. A mixture of a point mass at zero and a continuous distribution is used as the prior distribution on regression coefficients. In addition to the usual posterior, the fractional posterior, which is obtained by applying Bayes theorem with a fractional power of the likelihood, is also considered. The latter allows uniformity in posterior contraction over a larger subset of the parameter space. In our set-up, the link function of the generalized linear model need not be canonical. We show that Bayesian methods achieve convergence properties analogous to lasso-type procedures. Our results can be used to derive posterior contraction rates in many generalized linear models including logistic, Poisson regression and others.

SummaryPenalization of the likelihood by Jeffreys’ invariant prior, or a positive power thereof, is shown to produce finite-valued maximum penalized likelihood estimates in a broad class of binomial generalized linear models. The class of models includes logistic regression, where the Jeffreys-prior penalty is known additionally to reduce the asymptotic bias of the maximum likelihood estimator, and models with other commonly used link functions, such as probit and log-log. Shrinkage towards equiprobability across observations, relative to the maximum likelihood estimator, is established theoretically and studied through illustrative examples. Some implications of finiteness and shrinkage for inference are discussed, particularly when inference is based on Wald-type procedures. A widely applicable procedure is developed for computation of maximum penalized likelihood estimates, by using repeated maximum likelihood fits with iteratively adjusted binomial responses and totals. These theoretical results and methods underpin the increasingly widespread use of reduced-bias and similarly penalized binomial regression models in many applied fields.

SummaryWe propose a general framework based on selectively traversed accumulation rules for interactive multiple testing with generic structural constraints on the rejection set. It combines accumulation tests from ordered multiple testing with data-carving ideas from post-selection inference, allowing highly flexible adaptation to generic structural information. Our procedure defines an interactive protocol for gradually pruning a candidate rejection set, beginning with the set of all hypotheses and shrinking the set with each step. By restricting the information at each step via a technique we call masking, our protocol enables interaction while controlling the false discovery rate in finite samples for any data-adaptive update rule that the analyst may choose. We suggest update rules for a variety of applications with complex structural constraints, demonstrate that selectively traversed accumulation rules perform well in problems ranging from convex region detection to false discovery rate control on directed acyclic graphs, and show how to extend the framework to regression problems where knockoff statistics are available in lieu of $p$-values.

SummarySketching is a probabilistic data compression technique that has been largely developed by the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a smaller surrogate dataset. Typically, inference proceeds on the compressed dataset. Sketching algorithms generally use random projections to compress the original dataset, and this stochastic generation process makes them amenable to statistical analysis. We argue that the sketched data can be modelled as a random sample, thus placing this family of data compression methods firmly within an inferential framework. In particular, we focus on the Gaussian, Hadamard and Clarkson–Woodruff sketches and their use in single-pass sketching algorithms for linear regression with huge samples. We explore the statistical properties of sketched regression algorithms and derive new distributional results for a large class of sketching estimators. A key result is a conditional central limit theorem for data-oblivious sketches. An important finding is that the best choice of sketching algorithm in terms of mean squared error is related to the signal-to-noise ratio in the source dataset. Finally, we demonstrate the theory and the limits of its applicability on two datasets.

Summary We propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold, but is observable only in an infinite-dimensional space. Contamination of the predictor due to discrete or noisy measurements is also accounted for. By using functional local linear manifold smoothing, the proposed estimator enjoys a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the contamination level. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We also observe a phase transition phenomenon related to the interplay between the manifold dimension and the contamination level. We demonstrate via simulated and real data examples that the proposed method has favourable numerical performance relative to existing commonly used methods.

Summary Classifiers label data as belonging to one of a set of groups based on input features. It is challenging to achieve accurate classification when the feature distributions in the different classes are complex, with nonlinear, overlapping and intersecting supports. This is particularly true when training data are limited. To address this problem, we propose a new type of classifier based on obtaining a local approximation to the support of the data within each class in a neighbourhood of the feature to be classified, and assigning the feature to the class having the closest support. This general algorithm is referred to as local manifold approximation classification. As a simple and theoretically supported special case, which is shown to have excellent performance across a broad variety of examples, we use spheres for local approximation, obtaining a spherical approximation classifier.

Quantitative comparison of microbial composition from different populations is a fundamental task in various microbiome studies. We consider two-sample testing for microbial compositional data by leveraging phylogenetic information. Motivated by existing phylogenetic distances, we take a minimum-cost flow perspective to study such testing problems. We first show that multivariate analysis of variance with permutation using phylogenetic distances, one of the most commonly used methods in practice, is essentially a sum-of-squares type of test and has better power for dense alternatives. However, empirical evidence from real datasets suggests that the phylogenetic microbial composition difference between two populations is usually sparse. Motivated by this observation, we propose a new maximum type test, detector of active flow on a tree, and investigate its properties. We show that the proposed method is particularly powerful against sparse phylogenetic composition difference and enjoys certain optimality. The practical merit of the proposed method is demonstrated by simulation studies and an application to a human intestinal biopsy microbiome dataset on patients with ulcerative colitis.

Summary We consider an extension of Aalen’s additive regression model that allows covariates to have effects that vary on two different time scales. The two time scales considered are equal up to a constant for each individual and vary across individuals, such as follow-up time and age in medical studies or calendar time and age in longitudinal studies. The model was introduced in Scheike (2001), where it was solved using smoothing techniques. We present a new backfitting algorithm for estimating the structured model without having to use smoothing. Estimators of the cumulative regression functions on the two time scales are suggested by solving local estimating equations jointly on the two time scales. We provide large-sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction, providing a separation of the two effects stemming from time since diagnosis and age.