Optimal Estimation and Prediction for Dense Signals in High-Dimensional Linear Models

Estimation and prediction problems for dense signals are often framed in terms of minimax problems over highly symmetric parameter spaces. In this paper, we study minimax problems over l2-balls for high-dimensional linear models with Gaussian predictors. We obtain sharp asymptotics for the minimax risk that are applicable in any asymptotic setting where the number of predictors diverges and prove that ridge regression is asymptotically minimax. Adaptive asymptotic minimax ridge estimators are also identified. Orthogonal invariance is heavily exploited throughout the paper and, beyond serving as a technical tool, provides additional insight into the problems considered here. Most of our results follow from an apparently novel analysis of an equivalent non-Gaussian sequence model with orthogonally invariant errors. As with many dense estimation and prediction problems, the minimax risk studied here has rate d/n, where d is the number of predictors and n is the number of observations; however, when d is roughly proportional to n the minimax risk is influenced by the spectral distribution of the predictors and is notably different from the linear minimax risk for the Gaussian sequence model (Pinsker, 1980) that often appears in other dense estimation and prediction problems.Comment: 29 pages, 0 figure

Systematic Exclusion of Cognizable Groups by Use of Peremptory Challenges

An integral part of the jury selection process is the individual challenge, where a party has the privilege to remove potentially biased jurors. There are two forms of the individual challenge: for cause and peremptory. For cause challenges must be based on a legally provable bias, whereas peremptory challenges may be used to remove jurors who possess a bias peculiar to the immediate case, but whose bias is not articulable in terms of a challenge for cause. Given the vague underpinnings and potential abuses of the peremptory challenge, legislatures enforce a limitation on the number of such challenges a party may make. Nonetheless, peremptory challenges can still lead to systematic dismissals based solely on a juror\u27s affiliation with certain, cognizable groups, such as race, sex, ethnicity, and religion. This action undermines the integrity of the sixth amendment\u27s guarantee that juries will represent a fair cross-section of the community. This Note recommends that courts take greater initiative to question a party\u27s motives when it appears that peremptories have been applied discriminatorily

Engulfed by the Spectrum: The Impact of Autism Spectrum Disorders on Law and Policy

The recent increased awareness of Autism Spectral Disorders (“ASD”) has led to rapid development in laws meant to ensure disabled children receive adequate treatment and education. This development is due in large part to the proliferation of litigation initiated by individual ASD advocates, especially parents. Leaving the development of disability laws up to litigation, however, has resulted in often uneven and inconsistent interpretations of federal disability legislation. Moreover, the push to interpret these laws as they relate to children with ASD and their families has engulfed the development of disability law in general, especially where special education is concerned. This Article calls for replacing the current model of isolated parties advocating for their own individual interests with a more targeted, planned, and cohesive advocacy effort that includes parents, doctors, attorneys, and other professionals working together toward a common goal. To reach this conclusion, this Article first provides an introduction to ASD and its treatment and then examines the history of special education law in the United States, with particular attention paid to the Individuals with Disabilities Education Act. The Article then traces the evolution of ASD litigation in the courts, analyzing the major trends that have emerged involving diagnosis, treatment, placement of children within the education system, and funding of services. Finally, the Article surveys the various recent developments in federal and state legislation, focusing specially on insurance coverage requirements, federal health care reform, funding, education practices, and oversight mechanisms in legislation at the state level. In doing so, the Article reveals the need for a more cohesive and comprehensive approach to ensuring that law and policy related to ASD reflect the true needs of children and adults with ASD