We introduce the concept of coverage risk as an error measure for density ridge estimation.The coverage risk generalizes the mean integrated square error to set estimation.We propose two risk estimators for the coverage risk and we show that we can select tuning parameters by minimizing the estimated risk.We study the rate of convergence for coverage risk and prove consistency of the risk estimators.We apply our method to three simulated datasets and to cosmology data.In all the examples, the proposed method successfully recover the underlying density structure.

We present an efficient algorithm to actively select queries for learning the boundaries separating a function domain into regions where the func- tion is above and below a given threshold. We develop experiment selec- tion methods based on entropy, misclassification rate, variance, and their combinations, and show how they perform on a number of data sets. We then show how these algorithms are used to determine simultaneously valid 1 − α confidence intervals for seven cosmological parameters. Ex- perimentation shows that the algorithm reduces the computation neces- sary for the parameter estimation problem by an order of magnitude.

Performance attribution is usually conducted under the linear framework of multifactor models. Although commonly used by practitioners in finance, linear multifactor models are known to be less than satisfactory in many situations. After a brief survey of nonlinear methods, nonlinear statistical techniques are applied to performance attribution of a portfolio constructed from a fixed universe of stocks using factors derived from some commonly used cross sectional linear multifactor models. By rebalancing this portfolio monthly, the cumulative returns for procedures based on standard linear multifactor model and three nonlinear techniques-model selection, additive models, and neural networks-are calculated and compared. It is found that the first two nonlinear techniques, especially in combination, outperform the standard linear model. The results in the neural-network case are inconclusive because of the great variety of possible models. Although these methods are more complicated and may require some tuning, toolboxes are developed and suggestions on calibration are proposed. This paper demonstrates the usefulness of modern nonlinear statistical techniques in performance attribution.