Quantile regression trees, statistical applications of CUDA programming and identification of active effects without sparsity assumption

We introduce binary regression tree methods for estimating quantiles. Quantile regression trees can capture the relationship between the response and the predictors at different quantiles of the distribution. The trees are constructed by recursively partitioning the predictor space into terminal nodes and fitting linear quantile regression models to the terminal nodes. We provide two options to choose the partition: (1) search over all the predictors and all possible splits to choose the predictor and split that minimizes the quantile loss; (2) use a chi2 test to choose the split predictor and search over the splits of that predictor to find the split point. We also propose a tree method that overcomes the problem of crossing quantiles. A cube root rate of convergence of the split point is derived. We illustrate our tree methods with examples on Central America weather data and American infant birthweight data. Comparisons of our methods with some other quantile regression methods are given.;Many statistical applications can be implemented in a parallel way. On a single computer, the parallel computing can be conducted in the CUDA programming model on the graphics processing unit (GPU). We introduce some statistical methods that can be implemented in CUDA. Running the code in CUDA on GPU shows superior computing power compared to running on central processing unit (CPU). In local polynomial example, we gain about 100 times speed up compared to the CPU execution.;Most active effect identification methods for unreplicated factorial design require the sparsity assumption, that only a small fraction of the effects are actually active. The performance of many methods goes down when the number of active effects increases. We propose to combine a normality test statistic and another statistic derived from Dong's method into a step-down testing procedure. This approach does not need to assume sparsity. Our simulation results show that it can obtain consistent performance even when there are a large number of active effects.