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Frequency domain methods form a ubiquitous part of the statistical toolbox for time series analysis. In recent years, considerable interest has been given to the development of new spectral methodology and tools capturing dynamics in the entire joint distributions and thus avoiding the limitations of classical, $L^2$-based spectral methods. Most of the spectral concepts proposed in that literature suffer from one major drawback, though: their estimation requires the choice of a smoothing parameter, which has a considerable impact on estimation quality and poses challenges for statistical inference. In this paper, associated with the concept of copula-based spectrum, we introduce the notion of copula spectral distribution function or integrated copula spectrum. This integrated copula spectrum retains the advantages of copula-based spectra but can be estimated without the need for smoothing parameters. We provide such estimators, along with a thorough theoretical analysis, based on a functional central limit theorem, of their asymptotic properties. We leverage these results to test various hypotheses that cannot be addressed by classical spectral methods, such as the lack of time-reversibility or asymmetry in tail dynamics., 74 pages, 28 figures

Consider a panel data setting where repeated observations on individuals are available. Often it is reasonable to assume that there exist groups of individuals that share similar effects of observed characteristics, but the grouping is typically unknown in advance. We propose a novel approach to estimate such unobserved groupings for general panel data models. Our method explicitly accounts for the uncertainty in individual parameter estimates and remains computationally feasible with a large number of individuals and/or repeated measurements on each individual. The developed ideas can be applied even when individual-level data are not available and only parameter estimates together with some quantification of uncertainty are given to the researcher.

Instrumental variable methods provide useful tools for inferring causal effects in the presence of unmeasured confounding. To apply these methods with large-scale data sets, a major challenge is to find valid instruments from a possibly large candidate set. In practice, most of the candidate instruments are often not relevant for studying a particular exposure of interest. Moreover, not all relevant candidate instruments are valid as they may directly influence the outcome of interest. In this article, we propose a data-driven method for causal inference with many candidate instruments that addresses these two challenges simultaneously. A key component of our proposal is a novel resampling method, which constructs pseudo variables to remove irrelevant candidate instruments having spurious correlations with the exposure. Synthetic data analyses show that the proposed method performs favourably compared to existing methods. We apply our method to a Mendelian randomization study estimating the effect of obesity on health-related quality of life.

Extremal graphical models encode the conditional independence structure of multivariate extremes. For the popular class of H\"usler--Reiss models, we propose a majority voting algorithm for learning the underlying graph from data through $L^1$ regularized optimization. We derive explicit conditions that ensure consistent graph recovery for general connected graphs. A key statistic in our method is the empirical extremal variogram. We prove non-asymptotic concentration bounds for this quantity that hold for general multivariate Pareto distributions and are of independent interest., Comment: 70 pages, 12 figures. Previous title: "Concentration bounds for the extremal variogram"

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

Classical spectral analysis is based on the discrete Fourier transform of the auto-covariances. In this paper we investigate the asymptotic properties of new frequency domain methods where the auto-covariances in the spectral density are replaced by alternative dependence measures which can be estimated by U-statistics. An interesting example is given by Kendall{'}s $\tau$ , for which the limiting variance exhibits a surprising behavior.

This paper investigates the problem whether the difference between two parametric models $m_1,m_2$ describing the relation between a response variable and several covariates in two different groups is practically irrelevant, such that inference can be performed on the basis of the pooled sample. Statistical methodology is developed to test the hypotheses $H_0 : d(m_1,m_2)\geq \epsilon$ versus $H_1 : d(m_1,m_2) < \epsilon$ to demonstrate equivalence between the two regression curves $m_1,m_2$ for a pre-specified threshold $\epsilon$, where $d$ denotes a distance measuring the distance between $m_1$ and $m_2$. Our approach is based on the asymptotic properties of a suitable estimator $d(\hat{m}_1; \hat{m}_2)$ of this distance. In order to improve the approximation of the nominal level for small sample sizes a bootstrap test is developed, which addresses the specific form of the interval hypotheses. In particular, data has to be generated under the null hypothesis, which implicitly defines a manifold for the parameter vector. The results are illustrated by means of a simulation study and a data example. It is demonstrated that the new methods substantially improve currently available approaches with respect to power and approximation of the nominal level., Comment: 28 pages Keywords and Phrases: dose response studies; nonlinear regression; equivalence of curves; constrained parameter estimation; parametric bootstrap

In a recent paper Noh et al. (2013) proposed a new semiparametric estimate of a regression function with a multivariate predictor, which is based on a specification of the dependence structure between the predictor and the response by means of a parametric copula. This paper investigates the effect which occurs under misspecification of the parametric model. We demonstrate that even for a one or two dimensional predictor the error caused by a \wrong" specification of the parametric family is rather severe, if the regression is not monotone in one of the components of the predictor. Moreover, we also show that these problems occur for all of the commonly used copula families and we illustrate in several examples that the copula-based regression may lead to invalid results even when more exible copula models such as vine copulae (with the common parametric families) are used in the estimation procedure., Comment: Keywords: curse of dimensionality, semiparametric inference, copulae, pairwise copulae, vine copulae 14 pages, 8 figures