Mathematics

Subgraph counts - in particular the number of occurrences of small shapes such as triangles - characterize properties of random networks, and as a result have seen wide use as network summary statistics. However, subgraphs are typically counted globally, and existing approaches fail to describe vertex-specific characteristics. On the other hand, rooted subgraph counts - counts focusing on any given vertex's neighborhood - are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model which generalizes many popular statistical network models. This result enables a shift in the statistical analysis of large graphs, from estimating network summaries, to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that local friendship patterns are significant predictors of gender and race.

We obtain central limit theorems for stationary random fields employing a novel measure of dependence called θ -lex weak dependence. We show that this dependence notion is more general than strong mixing, i.e., it applies to a broader class of models. Moreover, we discuss hereditary properties for θ -lex and η -weak dependence and illustrate the possible applications of the weak dependence notions to the study of the asymptotic properties of stationary random fields. Our general results apply to mixed moving average fields (MMAF in short) and ambit fields. We show general conditions such that MMAF and ambit fields, with the volatility field being an MMAF or a p -dependent random field, are weakly dependent. For all the models mentioned above, we give a complete characterization of their weak dependence coefficients and sufficient conditions to obtain the asymptotic normality of their sample moments. Finally, we give explicit computations of the weak dependence coefficients of MSTOU processes and analyze under which conditions the developed asymptotic theory applies to CARMA fields.

We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.

We give two new simple characterizations of the Cauchy distribution by using the Möbius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.

A recent UK Biobank study clustered 156 parameterised models associating risk factors with common diseases, to identify shared causes of disease. Parametric models are often more familiar and interpretable than clustered data, can build-in prior knowledge, adjust for known confounders, and use marginalisation to emphasise parameters of interest. Estimates include a Maximum Likelihood Estimate (MLE) that is (approximately) normally distributed, and its covariance. Clustering models rarely consider the covariances of data points, that are usually unavailable. Here a clustering model is formulated that accounts for covariances of the data, and assumes that all MLEs in a cluster are the same. The log-likelihood is exactly calculated in terms of the fitted parameters, with the unknown cluster means removed by marginalisation. The procedure is equivalent to calculating the Bayesian Information Criterion (BIC) without approximation, and can be used to assess the optimum number of clusters for a given clustering algorithm. The log-likelihood has terms to penalise poor fits and model complexity, and can be maximised to determine the number and composition of clusters. Results can be similar to using the ad-hoc "elbow criterion", but are less subjective. The model is also formulated as a Dirichlet process mixture model (DPMM). The overall approach is equivalent to a multi-layer algorithm that characterises features through the normally distributed MLEs of a fitted model, and then clusters the normal distributions. Examples include simulated data, and clustering of diseases in UK Biobank data using estimated associations with risk factors. The results can be applied directly to measured data and their estimated covariances, to the output from clustering models, or the DPMM implementation can be used to cluster fitted models directly.

To perform statistical inference for time series, one should be able to assess if they present deterministic or stochastic trends. For univariate analysis one way to detect stochastic trends is to test if the series has unit roots, and for multivariate studies it is often relevant to search for stationary linear relationships between the series, or if they cointegrate. The main goal of this article is to briefly review the shortcomings of unit root and cointegration tests proposed by the Bayesian approach of statistical inference and to show how they can be overcome by the fully Bayesian significance test (FBST), a procedure designed to test sharp or precise hypothesis. We will compare its performance with the most used frequentist alternatives, namely, the Augmented Dickey-Fuller for unit roots and the maximum eigenvalue test for cointegration. Keywords: Time series; Bayesian inference; Hypothesis testing; Unit root; Cointegration.

We consider MCMC methods for learning equivalence classes of sparse Gaussian DAG models when p= e o(n) . The main contribution of this work is a rapid mixing result for a random walk Metropolis-Hastings algorithm, which we prove using a canonical path method. It reveals that the complexity of Bayesian learning of sparse equivalence classes grows only polynomially in n and p , under some common high-dimensional assumptions. Further, a series of high-dimensional consistency results is obtained by the path method, including the strong selection consistency of an empirical Bayes model for structure learning and the consistency of a greedy local search on the restricted search space. Rapid mixing and slow mixing results for other structure-learning MCMC methods are also derived. Our path method and mixing time results yield crucial insights into the computational aspects of high-dimensional structure learning, which may be used to develop more efficient MCMC algorithms.

Algorithms are proposed for the computation of set-valued quantiles and the values of the lower cone distribution function for bivariate data sets. These new objects make data analysis possible involving an order relation for the data points in form of a vector order in two dimensions. The bivariate case deserves special attention since two-dimensional vector orders are much simpler to handle than such orders in higher dimensions. Several examples illustrate how the algorithms work and what kind of conclusions can be drawn with the proposed approach.

One fundamental goal of high-dimensional statistics is to detect or recover structure from noisy data. In many cases, the data can be faithfully modeled by a planted structure (such as a low-rank matrix) perturbed by random noise. But even for these simple models, the computational complexity of estimation is sometimes poorly understood. A growing body of work studies low-degree polynomials as a proxy for computational complexity: it has been demonstrated in various settings that low-degree polynomials of the data can match the statistical performance of the best known polynomial-time algorithms for detection. While prior work has studied the power of low-degree polynomials for the task of detecting the presence of hidden structures, it has failed to address the estimation problem in settings where detection is qualitatively easier than estimation. In this work, we extend the method of low-degree polynomials to address problems of estimation and recovery. For a large class of "signal plus noise" problems, we give a user-friendly lower bound for the best possible mean squared error achievable by any degree-D polynomial. To our knowledge, this is the first instance in which the low-degree polynomial method can establish low-degree hardness of recovery problems where the associated detection problem is easy. As applications, we give a tight characterization of the low-degree minimum mean squared error for the planted submatrix and planted dense subgraph problems, resolving (in the low-degree framework) open problems about the computational complexity of recovery in both cases.

We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises, for example, testing the significance of X in a scalar-on-function linear regression model of response Y on functional regressors X and Z. We show however that even in the idealised setting where additionally (X, Y, Z) have a non-singular Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed to restrict the null and we argue that a convenient way of specifying these is based on choosing methods for regressing each of X and Y on Z. We thus propose as a test statistic, the Hilbert-Schmidt norm of the outer product of the resulting residuals, and prove that type I error control is guaranteed when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models.