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Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science and applied mathematics. In this paper we characterize the behaviour of perturbed eigenvectors for a range of signal-plus-noise matrix models encountered in statistical and randommatrix-theoretic settings. We establish both first-order approximation results, i.e., sharp deviations, and second-order distributional limit theory, i.e., fluctuations. The concise methodology presented in this paper synthesizes tools rooted in two core concepts, namely deterministic decompositions of matrix perturbations and probabilistic matrix concentration phenomena. We illustrate our theoretical results with simulation examples involving stochastic block model random graphs.

Many causal questions involve interactions between units, also known as interference, for example between individuals in households, students in schools, or firms in markets. In this paper we formalize the concept of a conditioning mechanism, which provides a framework for constructing valid and powerful randomization tests under general forms of interference. We describe our framework in the context of two-stage randomized designs and apply our approach to a randomized evaluation of an intervention targeting student absenteeism in the school district of Philadelphia. We show improvements over existing methods in terms of computational and statistical power.

Strong orthogonal arrays have better space-filling properties than ordinary orthogonal arrays for computer experiments. We consider column-orthogonal strong orthogonal arrays of strength two plus and three minus, and present methods of constructing such designs. Several situations are examined, including those of four or higher levels and mixed levels. The methods are based on both regular and nonregular designs. The resulting designs inherit the good property of strong orthogonal arrays of strength two plus or three and have the additional property of column orthogonality. This type of design is a better choice for computer experiments.

Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappé et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm.

In paired randomized experiments, individuals in a given matched pair may differ on prognostically important covariates despite the best efforts of practitioners. We examine the use of regression adjustment to correct for persistent covariate imbalances after randomization, and present two regression-assisted estimators for the sample average treatment effect in paired experiments. Using the potential outcomes framework, we prove that these estimators are consistent for the sample average treatment effect under mild regularity conditions even if the regression model is improperly specified, and describe how asymptotically conservative confidence intervals can be constructed. We demonstrate that the variances of the regression-assisted estimators are no larger than that of the standard difference-in-means estimator asymptotically, and illustrate the proposed methods by simulation. The analysis does not require a superpopulation model, a constant treatment effect, or the truth of the regression model, and hence provides inference for the sample average treatment effect with the potential to increase power without unrealistic assumptions.

FIVE BOOKS IN ONE! This collection includes the following 5 complete Ideas Roadshow books featuring leading researchers providing fully accessible insights into cutting-edge academic research while revealing the inspirations and personal journeys behind the research. A detailed preface highlights the connections between the different books and all five books are broken into chapters with a detailed introduction and questions for discussion at the end of each chapter: 1. The Power of Principles: Physics Revealed - A Conversation with Nima Arkani-Hamed, faculty member at the renowned Institute for Advanced Study in Princeton. Prof. Arkani-Hamed is one of today's leading particle physicists. This extensive Ideas Roadshow conversation explores how we discover the laws of nature, the \"scientific method\", the relation between theory and experiment and how we can push our understanding well beyond where experiments can currently reach.2. Cryptoreality - A Conversation with Artur Ekert, Professor of Quantum Physics at the Mathematical Institute at the University of Oxford and Director of the Centre for Quantum Technologies and Lee Kong Chian Centennial Professor at the National University of Singapore. Artur Ekert is one of the pioneers of quantum cryptography. This wide-ranging conversation provides detailed insights into his research and covers many fascinating topics such as mathematical and physical intuition, a detailed history of cryptography from antiquity to the present day and how it works in practice, the development of quantum information science, the nature of reality, and more.3. The Problems of Physics, Reconsidered - A Conversation with Physics Nobel Laureate Tony Leggett, University of Illinois. The basis of this conversation is Tony Leggett's book The Problems of Physics and further explores the insightful plain-speaking itemization that he developed of the physics landscape according to four basic categories—the very small (particle physics), the very large (cosmology), the very complex (condensed matter physics) and the very unclear (foundations of quantum theory)—while providing a thoughtful follow-up analysis from a contemporary perspective to assess how much progress we've made and which mysteries remain or have come on the scene since the book was published.4. The Physics of Banjos - A Conversation with David Politzer, 2004 Nobel Laureate and the Richard Chace Tolman Professor of Theoretical Physics at Caltech. This extensive conversation examines many of the intriguing aspects associated with the physics of banjos, including the ocarina effect, string-stretching, the subtleties of how we hear pitch, transient growth, and the mysterious ringing sound of banjos; while also touching briefly on contemporary issues in black holes and particle physics.5. Indiana Steinhardt and the Quest for Quasicrystals - A Conversation with Paul Steinhardt, the Albert Einstein Professor of Science and Director of the Center for Theoretical Science at Princeton University. This extensive conversation provides a comprehensive account of a marvellous scientific adventure story in the quest for a natural quasicrystal. The reader will be taken on a fascinating ride through the physics of materials, from theory, to the laboratory, to the discovery of a new state of matter, that culminated in Paul Steinhardt's dramatic Siberian expedition. Paul Steinhardt talks about his encounters with mineral smugglers, secret diaries and quasi-mythical characters during his \"Indiana Jones\" expedition from Florence to Israel, Amsterdam to California, Princeton to Kamchatka which led him to find quasicrystals that are quite literally out of this world…Howard Burton is the founder and host of all Ideas Roadshow Conversations and was the Founding Executive Director of Perimeter Institute for Theoretical Physics. He holds a PhD in theoretical physics and an MA in philosophy.

Maximin distance designs and orthogonal designs are widely used in computer and physical experiments. We characterize a broad class of maximin distance designs by establishing new bounds on the minimum intersite distance for mirror-symmetric and general U-type designs. We show that maximin distance designs and orthogonal designs are closely related and coincide under some conditions.