On robust estimation of the location parameter

KEGG(Kyoto Encyclopedia of Genes and Genomes)〔和文〕 (特集 ゲノム医学の現在と未来--基礎と臨床) -- (データベース)

Today's strongly connected, global networks have produced highly interdependent systems that we do not understand and cannot control well. These systems are vulnerable to failure at all scales, posing serious threats to society, even when external shocks are absent. As the complexity and interaction strengths in our networked world increase, man-made systems can become unstable, creating uncontrollable situations even when decision-makers are well-skilled, have all data and technology at their disposal, and do their best. To make these systems manageable, a fundamental redesign is needed. A 'Global Systems Science' might create the required knowledge and paradigm shift in thinking.

Globally networked risks and how to respond

The statistical analysis of compositional data

Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

The global financial crisis of 2007–2009 exposed critical weaknesses in the financial system. Many proposals for financial reform address the need for systemic regulation—that is, regulation focused on the soundness of the whole financial system and not just that of individual institutions. In this paper, we study one particular problem faced by a systemic regulator: the tension between the distribution of assets that individual banks would like to hold and the distribution across banks that best supports system stability if greater weight is given to avoiding multiple bank failures. By diversifying its risks, a bank lowers its own probability of failure. However, if many banks diversify their risks in similar ways, then the probability of multiple failures can increase. As more banks fail simultaneously, the economic disruption tends to increase disproportionately. We show that, in model systems, the expected systemic cost of multiple failures can be largely explained by two global parameters of risk exposure and diversity, which can be assessed in terms of the risk exposures of individual actors. This observation hints at the possibility of regulatory intervention to promote systemic stability by incentivizing a more diverse diversification among banks. Such intervention offers the prospect of an additional lever in the armory of regulators, potentially allowing some combination of improved system stability and reduced need for additional capital.

/pdf/individual-versus-systemic-risk-and-the-regulator-s-dilemma-54e4wsmo6s.pdf

Individual versus systemic risk and the Regulator's Dilemma

This paper studies hypothesis testing and parameter estimation in the context of the divide-and-conquer algorithm. In a unified likelihood based framework, we propose new test statistics and point estimators obtained by aggregating various statistics from k subsamples of size n/k, where n is the sample size. In both low dimensional and sparse high dimensional settings, we address the important question of how large k can be, as n grows large, such that the loss of efficiency due to the divide-and-conquer algorithm is negligible. In other words, the resulting estimators have the same inferential efficiencies and estimation rates as an oracle with access to the full sample. Thorough numerical results are provided to back up the theory.

/pdf/distributed-testing-and-estimation-under-sparse-high-36pc5lozm7.pdf

Distributed testing and estimation under sparse high dimensional models.

This paper studies hypothesis testing and parameter estimation in the context of the divide and conquer algorithm. In a unied likelihood based framework, we propose new test statistics and point estimators obtained by aggregating various statistics from k subsamples of size n=k, where n is the sample size. In both low dimensional and high dimensional settings, we address the important question of how to choose k as n grows large, providing a theoretical upper bound on k such that the information loss due to the divide and conquer algorithm is negligible. In other words, the resulting estimators have the same inferential eciencies and estimation rates as a practically infeasible oracle with access to the full sample. Thorough numerical results are provided to back up the theory.

Distributed Estimation and Inference with Statistical Guarantees

High-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded 2 + e moments for e ∈ (0, 2). The associated convergence rates depend on e.

Robust estimation of high-dimensional covariance and precision matrices

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.

/pdf/a-topologically-valid-definition-of-depth-for-functional-zzji4jqhwt.pdf

Heather Battey

Papers

Individual versus systemic risk and the Regulator's Dilemma

Distributed testing and estimation under sparse high dimensional models.

Distributed Estimation and Inference with Statistical Guarantees

Robust estimation of high-dimensional covariance and precision matrices

A Topologically Valid Definition of Depth for Functional Data