Regression Quantiles Computation and Applications

Portnoy 9703758：传统统计线性模型试图用预测变量（如教育、职业培训、社会特征等）来解释响应变量（如工资）。经典的方法使用“租赁平方”来估计响应的平均值，条件取决于预测因子。然而，人们经常发现，那些反应较高的人对预测因子的依赖程度与反应中等或较低的人非常不同。经典方法完全失去了这种可变性。因此，现代回归分位数方法越来越受欢迎。这些方法试图根据预测因子估计反应的条件分位数（百分位数）。例如，人们发现，高工资比低工资更强烈地依赖于教育。也就是说，高工资不仅与更多的教育相关，而且高工资人群的教育回报率也明显高于低工资人群。同样，夏季用电高峰的人白天用电高峰和夜间用电高峰之间的差异要比用电高峰的人大得多（可能是因为空调的缘故），长期住院的时间长短在很大程度上取决于疾病的严重程度，而不是短期住院的时间长短。这种根据响应大小区分模型的能力在经济学、社会科学、生物统计学和其他领域得到了广泛的应用。不可避免地，这些方法的成功持续传播与方便和高效软件的可用性密切相关。对于中等规模的问题，现有的算法需要与最小二乘相当的计算量。然而，随着问题规模的增长，以前的方法的计算负担变得沉重。对于经验劳动经济学、大型生物医学调查和其他领域中常见的大问题，新的、更有效的计算技术将是非常可取的。因此，最近的研究提出了一种双管齐下的方法，这种方法已被证明可以显著提高计算效率。一个方面是使用最近发展的内点法进行线性规划。第二种是随机预处理的一种形式，它可以大大减少大多数统计回归分位数问题的有效大小。总之，这些想法似乎能够将计算工作降低到最小二乘的水平，以解决样本量高达数百万个观察值的问题。在更大的问题中，理论证据表明回归分位数方法甚至比最小二乘法更快。这一理论改进的实际实现将对大型和海量数据集的高性能计算领域产生重大影响。

Portnoy 9703758: Traditional Statistical Linear Modeling seeks to explain a response variable (e.g., wages) in terms of predictor variables (e.g., education, job training, social characteristics, etc.). The classical approach uses "lease squares" to estimate the mean of the responses conditional on the predictors. It is often found, however, that those with higher responses depend very differently on the predictors than those with middle or lower responses. This variability is completely lost by the classical approach. Thus, modern regression quantile methods have become increasingly popular. These methods seek to estimate the conditional quantiles (percentiles) of the response in terms of the predictors. For example, it has been found that high wages depend much more strongly on education than lower wages. That is, not only are high wages associated with more education, but the rate of return on education is significantly higher for high wage earners than for lower wage earners. Similarly, high electricity consumers during the summer show a much greater difference between daytime peak use and nighttime use than lower users (presumably because of air conditioners), and the length of long hospital stays depends more strongly on the severity of the disease than does the length of shorter stays. This ability to distinguish models on the basis of the size of the response is finding extensive application in economics, social sciences, biostatistics, and other areas. Inevitably, successful continued diffusion of these methods is linked closely to the availability of convenient and efficient software. For modestly large problems, existing algorithms require computational effort comparable to least squares. However, as problem size grows, the computational burden of the previous methods becomes heavy. For large problems common in empirical labor economics, large biomedical surveys, and other areas, new and more efficient computational techniques would be highly des irable. Thus, recent research proposes a two-pronged attack that has been shown to yield dramatic improvements in computational efficiency. One prong is the use of recently developed interior point methods for linear programming. The second is a form of stochastic preprocessing which can drastically reduce the effective size of most statistical regression quantile problems. Together, these ideas appear capable of bringing the computational effort down to the level of least squares for problems with sample sizes up to several million observations. In even larger problems, theoretical evidence indicates that regression quantile methods are even faster than least squares. Practical realization of this theoretical improvement would have significant consequences in the area of high performance computation for large and massive data sets.