Distributional approximations of (bi-) linear functions of sample variancecovariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension d is allowed to grow with the sample size n and may even be larger than n. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly ℓ 1 -bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high-dimensional time series, which matter for the analysis of big data and are discussed in greater detail.
The research on and application of artificial intelligence (AI) has triggered a comprehensive scientific, economic, social and political discussion. Here we argue that statistics, as an interdisciplinary scientific field, plays a substantial role both for the theoretical and practical understanding of AI and for its future development. Statistics might even be considered a core element of AI. With its specialist knowledge of data evaluation, starting with the precise formulation of the research question and passing through a study design stage on to analysis and interpretation of the results, statistics is a natural partner for other disciplines in teaching, research and practice. This paper aims at highlighting the relevance of statistical methodology in the context of AI development. In particular, we discuss contributions of statistics to the field of artificial intelligence concerning methodological development, planning and design of studies, assessment of data quality and data collection, differentiation of causality and associations and assessment of uncertainty in results. Moreover, the paper also discusses the equally necessary and meaningful extensions of curricula in schools and universities to integrate statistical aspects into AI teaching.
Motivated in part by applications in model selection in statistical genetics and sequential monitoring of financial data, we study an empirical process framework for a class of stopping rules which rely on kernel-weighted averages of past data. We are interested in the asymptotic distribution for time series data and an analysis of the joint influence of the smoothing policy and the alternative defining the deviation from the null model (in-control state). We employ a certain type of local alternative which provides meaningful insights. Our results hold true for short memory processes which satisfy a weak mixing condition. By relying on an empirical process framework we obtain both asymptotic laws for the classical fixed sample design and the sequential monitoring design. As a by-product we establish the asymptotic distribution of the Nadaraya-Watson kernel smoother when the regressors do not get dense as the sample size increases. Copyright Springer-Verlag 2004Control chart, Finance, Microarrays, Sequential test, Smoothing, Statistical genetics,
Abstract. Motivated by applications in statistical quality control and signal analysis, we propose a sequential detection procedure which is designed to detect structural changes, in particular jumps, immediately. This is achieved by modifying a median filter by appropriate kernel-based jump preserving weights (shrinking) and a clipping mechanism. We aim at both robustness and immediate detection of jumps. Whereas the median approach ensures robust smooths when there are no jumps, the modification ensure immediate reaction to jumps. For general clipping location estimators we show that the procedure can detect jumps of certain heights with no delay, even when applied to Banach space valued data. For shrinking medians we provide an asymptotic upper bound for the normed delay. The finite sample properties are studied by simulations which show that our proposal outperforms classical procedures in certain respects.
We establish large sample approximations for an arbitray number of bilinear forms of the sample variance-covariance matrix of a high-dimensional vector time series using ℓ1-bounded weighting vectors. Estimation of the asymptotic covariance structure is also discussed. The results hold true without any constraint on the dimension, the number of forms and the sample size or their ratios. Concrete and potential applications are widespread and cover highdimensional data science problems such as projections onto sparse principal components or more general spanning sets as frequently considered, e.g. in classification and dictionary learning. As two specific applications of our results, we study in greater detail the asymptotics of the trace functional and shrinkage estimation of the covariance matrices. In shrinkage estimation, it turns out that the asymptotics differs for weighting vectors bounded away from orthogonaliy and nearly orthogonal ones in the sense that their inner product converges to 0.
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