Validating Approximate Slope Homogeneity in Large Panels

Statistical inference for large data panels is omnipresent in modern economic applications. An important benefit of panel analysis is the possibility to reduce noise and thus to guarantee stable inference by intersectional pooling. However, it is wellknown that pooling can lead to a biased analysis if individual heterogeneity is too strong. In classical linear panel models, this trade-off concerns the homogeneity of slope parameters, and a large body of tests has been developed to validate this assumption. Yet, such tests can detect inconsiderable deviations from slope homogeneity, discouraging pooling, even when practically beneficial. In order to permit a more pragmatic analysis, which allows pooling when individual heterogeneity is sufficiently small, we present in this paper the concept of approximate slope homogeneity. We develop an asymptotic level $\alpha$ test for this hypothesis, that is uniformly consistent against classes of local alternatives. In contrast to existing methods, which focus on exact slope homogeneity and are usually sensitive to dependence in the data, the proposed test statistic is (asymptotically) pivotal and applicable under simultaneous intersectional and temporal dependence. Moreover, it can accommodate the realistic case of panels with large intersections. A simulation study and a data example underline the usefulness of our approach

Lower Bounds for R\'enyi Differential Privacy in a Black-Box Setting

We present new methods for assessing the privacy guarantees of an algorithm with regard to R\'enyi Differential Privacy. To the best of our knowledge, this work is the first to address this problem in a black-box scenario, where only algorithmic outputs are available. To quantify privacy leakage, we devise a new estimator for the R\'enyi divergence of a pair of output distributions. This estimator is transformed into a statistical lower bound that is proven to hold for large samples with high probability. Our method is applicable for a broad class of algorithms, including many well-known examples from the privacy literature. We demonstrate the effectiveness of our approach by experiments encompassing algorithms and privacy enhancing methods that have not been considered in related works

Detecting structural breaks in eigensystems of functional time series

Detecting structural changes in functional data is a prominent topic in statistical literature. However not all trends in the data are important in applications, but only those of large enough in uence. In this paper we address the problem of identifying relevant changes in the eigenfunctions and eigenvalues of covariance kernels of L^2[0; 1]- valued time series. By self-normalization techniques we derive pivotal, asymptotically consistent tests for relevant changes in these characteristics of the second order structure and investigate their finite sample properties in a simulation study. The applicability of our approach is demonstrated analyzing German annual temperature data

Pivotal goodness-of-fit measures in functional data analysis

Die vorliegende Arbeit behandelt statistische Inferenz für funktionale Zeitreihen. Der Fokus liegt dabei auf der Schätzung und statistischen Quantifizierung von Maßen der Anpassungsgüte. Diese Maße werden unter der "klassischen Alternative" betrachtet (d. h. für nicht ideale Modellanpassung) und mithilfe von Konfidenzintervallen sowie einseitigen Hypothesentests untersucht. Die vorgestellte Methodologie hat sowohl Anwendungen für Ein- und Zweistichprobenprobleme, sowie für relevante Strukturbruchanalysen. Eine Besonderheit der präsentierten Methodik ist die Konstruktion "selbstnormalisierter Statistiken" mit verteilungsfreiem Grenzwert. Im Gegensatz zu alternativen Inferenzmethoden für abhängige Daten (wie z.B. Bootstrap oder die Schätzung der asymptotischen Varianz) sind selbstnormalisierte Statistiken wenig rechenintensiv und setzen außerdem keine Wahl von Bandbreiteparametern voraus, die ein Vorwissen über die Stärke der Abhängigkeit erfordern

Quantifying deviations from separability in space-time functional processes

The estimation of covariance operators of spatio-temporal data is in many applications only computationally feasible under simplifying assumptions, such as separability of the covariance into strictly temporal and spatial factors. Powerful tests for this assumption have been proposed in the literature. However, as real world systems, such as climate data are notoriously inseparable, validating this assumption by statistical tests, seems inherently questionable. In this paper we present an alternative approach: By virtue of separability measures, we quantify how strongly the data’s covariance operator diverges from a separable approximation. Confidence intervals localize these measures with statistical guarantees. This method provides users with a flexible tool, to weigh the computational gains of a separable model against the associated increase in bias. As separable approximations we consider the established methods of partial traces and partial products, and develop weak convergence principles for the corresponding estimators. Moreover, we also prove such results for estimators of optimal, separable approximations, which are arguably of most interest in applications. In particular we present for the first time statistical inference for this object, which has been confined to estimation previously. Besides confidence intervals, our results encompass tests for approximate separability. All methods proposed in this paper are free of nuisance parameters and do neither require computationally expensive resampling procedures nor the estimation of nuisance parameters. A simulation study underlines the advantages of our approach and its applicability is demonstrated by the investigation of German annual temperature data

The empirical process of residuals from an inverse regression

In this paper we investigate an indirect regression model characterized by the Radon transformation. This model is useful for recovery of medical images obtained by computed tomography scans. The indirect regression function is estimated using a series estimator motivated by a spectral cut-off technique. Further, we investigate the empirical process of residuals from this regression, and show that it satsifies a functional central limit theorem