Asymptotic inference for high-dimensional data

In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is, allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide precise conditions for the joint consistency of the estimators of the mean. In the process, we clarify and improve some of the recent consistency results that appeared in the literature. An important aspect of the work presented is the development of asymptotic normality results for these models. As a consequence, we construct different test statistics for one-sample and two-sample problems concerning the mean vector and obtain their asymptotic distributions as a corollary of the infinite-dimensional results. Finally, we use these theoretical results to develop an asymptotically justifiable methodology for data analyses. Simulation results presented here describe situations where the methodology can be successfully applied. They also evaluate its robustness under a variety of conditions, some of which are substantially different from the technical conditions. Comparisons to other methods used in the literature are provided. Analyses of real-life data is also included.Comment: Published in at http://dx.doi.org/10.1214/09-AOS718 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes''

Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes'' [math.PR/0407059]Comment: Published at http://dx.doi.org/10.1214/105051606000000574 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local limit theory and large deviations for supercritical Branching processes

In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n

An Efficient Bandit Algorithm for Realtime Multivariate Optimization

Optimization is commonly employed to determine the content of web pages, such as to maximize conversions on landing pages or click-through rates on search engine result pages. Often the layout of these pages can be decoupled into several separate decisions. For example, the composition of a landing page may involve deciding which image to show, which wording to use, what color background to display, etc. Such optimization is a combinatorial problem over an exponentially large decision space. Randomized experiments do not scale well to this setting, and therefore, in practice, one is typically limited to optimizing a single aspect of a web page at a time. This represents a missed opportunity in both the speed of experimentation and the exploitation of possible interactions between layout decisions. Here we focus on multivariate optimization of interactive web pages. We formulate an approach where the possible interactions between different components of the page are modeled explicitly. We apply bandit methodology to explore the layout space efficiently and use hill-climbing to select optimal content in realtime. Our algorithm also extends to contextualization and personalization of layout selection. Simulation results show the suitability of our approach to large decision spaces with strong interactions between content. We further apply our algorithm to optimize a message that promotes adoption of an Amazon service. After only a single week of online optimization, we saw a 21% conversion increase compared to the median layout. Our technique is currently being deployed to optimize content across several locations at Amazon.com.Comment: KDD'17 Audience Appreciation Awar

Astrometry and Photometry with Coronagraphs

We propose a solution to the problem of astrometric and photometric calibration of coronagraphic images with a simple optical device which, in theory, is easy to use. Our design uses the Fraunhofer approximation of Fourier optics. Placing a periodic grid of wires (we use a square grid) with known width and spacing in a pupil plane in front of the occulting coronagraphic focal plane mask produces fiducial images of the obscured star at known locations relative to the star. We also derive the intensity of these fiducial images in the coronagraphic image. These calibrator images can be used for precise relative astrometry, to establish companionship of other objects in the field of view through measurement of common proper motion or common parallax, to determine orbits, and to observe disk structure around the star quantitatively. The calibrator spots also have known brightness, selectable by the coronagraph designer, permitting accurate relative photometry in the coronagraphic image. This technique, which enables precision exoplanetary science, is relevant to future coronagraphic instruments, and is particularly useful for `extreme' adaptive optics and space-based coronagraphy.Comment: To appear in ApJ August 2006, 27 preprint style pages 4 figure