Optimal sampling strategies for multiscale stochastic processes

In this paper, we determine which non-random sampling of fixed size gives the best linear predictor of the sum of a finite spatial population. We employ different multiscale superpopulation models and use the minimum mean-squared error as our optimality criterion. In multiscale superpopulation tree models, the leaves represent the units of the population, interior nodes represent partial sums of the population, and the root node represents the total sum of the population. We prove that the optimal sampling pattern varies dramatically with the correlation structure of the tree nodes. While uniform sampling is optimal for trees with ``positive correlation progression'', it provides the worst possible sampling with ``negative correlation progression.'' As an analysis tool, we introduce and study a class of independent innovations trees that are of interest in their own right. We derive a fast water-filling algorithm to determine the optimal sampling of the leaves to estimate the root of an independent innovations tree.Comment: Published at http://dx.doi.org/10.1214/074921706000000509 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Brief history of the Lehmann Symposia: Origins, goals and motivation

The idea of the Lehmann Symposia as platforms to encourage a revival of interest in fundamental questions in theoretical statistics, while keeping in focus issues that arise in contemporary interdisciplinary cutting-edge scientific problems, developed during a conversation that I had with Victor Perez Abreu during one of my visits to Centro de Investigaci\'{o}n en Matem\'{a}ticas (CIMAT) in Guanajuato, Mexico. Our goal was and has been to showcase relevant theoretical work to encourage young researchers and students to engage in such work. The First Lehmann Symposium on Optimality took place in May of 2002 at Centro de Investigaci\'{o}n en Matem\'{a}ticas in Guanajuato, Mexico. A brief account of the Symposium has appeared in Vol. 44 of the Institute of Mathematical Statistics series of Lecture Notes and Monographs. The volume also contains several works presented during the First Lehmann Symposium. All papers were refereed. The program and a picture of the participants can be found on-line at the website http://www.stat.rice.edu/lehmann/lst-Lehmann.html.Comment: Published at http://dx.doi.org/10.1214/074921706000000347 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiscale smoothing error models

Caption title.Includes bibliographical references (p. 9-10).Supported by the Air Force Office of Scientific Research. AFOSR-92-J-0002 Supported by the Office of Naval Research. N00014-91-J-1004 Supported by the Army Research Office. DAAL03-92-G-0115Mark R. Luettgen, Alan S. Willsky

Analysis, estimation and control for perturbed and singular systems and for systems subject to discrete events.

Annual technical report for grant AFOSR-88-0032.Investigators: Alan S. Willsky, George C. Verghese.Includes bibliographical references (p. [10]-[15]).Research supported by the AFOSR. AFOSR-88-003

Multiscale Riccati equations and a two-sweep algorithm for the optimal fusion of multiresolution data

Cover title.Includes bibliographical references (p. 48-49).Research supported by the National Science Foundation. ECS-8700903 Research supported by the Air Force Office of Scientific Research. AFOSR-88-0032 Research supported by the US Army Research Office. DAAL03-86-K-0171K.C. Chou, A.S. Willsky

Analysis, estimation and control for perturbed and singular systems for systems subject to discrete events.

"The principle investigator for this effort is Professor Alan S. Willsky, and Professor George C. Verghese is co-principal investigator."--P. [3].Includes bibliographical references (p. [20]-[25]).Final technical report for grant AFOSR-88-0032.Supported by the AFOSR. AFOSR-88-003