基于分位回归的当代统计学逆问题重大基础理论和方法及其应用

With the rapid development of science and technology, it is common that high and ultra-high dimensional complex big data generates in many natural science and social science and interdisciplinary fields. Here an extremely challenging problem is that people can get data from a domain (usually from lower dimensional domain), but their interest is in another domain, this situation exists in many scientific fields, from medical imaging to physical chemistry to extragalactic astronomy. The contemporary statistical research has begun to pay attention to these problems, i.e., the statistical inverse problem. The project will construct a major basic theory of some inverse problems in contemporary statistics, propose new algorithms and perform large-scale actual data analysis. We consider a sequence of hierarchical space model of inverse problems. The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means. The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable, and an automatic selection principle for the nonrandom filters. This leads to the data-driven choice of weights. We also give an algorithm for its implementation. The quantile coupling inequality developed is of independent interest because it includes the median coupling inequality in literature as a special case..This subject has a high-tech flavor and is an international frontier research, and has great theoretical and practical significance.

随着科学技术迅猛发展，复杂高维乃至超高维大数据普遍产生于许多自然科学、社会科学及交叉学科领域。这里极富挑战的问题是人们能得到一个域中的数据(通常是较低维的域)，但我们的兴趣却是在另一个域中，这种情况在很多科学领域都存在，从医学影像到物理化学再到河外天文学等。当代统计研究开始关注这些问题，即统计逆（反）问题。该项目将构建当代统计学中的一些逆问题的重大基础理论，提出新算法与展开大规模的实际数据分析。考虑逆问题的高维空间模型序列，针对多种误差分布情况，通过非直接的观测值得到潜在函数的估计，研发分位耦合技术，建立逆问题的Oracle不等式。这些不等式给出任意样本的p-分位点和一个正态变量之间耦合程度的边界，以及非随机过滤的自动选择准则，促使基于数据驱动的权重选择，给出执行时的具体算法。这些分位耦合定理具有独立的学术价值。该项目极富高技术色彩，实属国际前沿研究，预期成果将具有重大理论价值和实际意义。

总体来说，该项目是按照计划执行的，有的地方甚至超额完成任务。该项目的研究目标已经达到,取得了一系列的重要结果。在包括国际国内顶级刊物上发表学术论文54篇，其中SCI文章20篇，其余的基本上都是核心期刊，出版专著1部，还有大量的文章在审稿中。有的发表在国际国内该领域的顶级刊物上，例如英国皇家统计学会杂志JRSS。下面仅列举10个方面的代表性研究结果：1) 竞争风险模型的高维变量选择建模理论方法;2)复杂高维分位逆回归下的变量选择建模理论方法;3)时空数据插值张量回归建模理论及方法;4)基于函数型数据的部分变系数混合建模理论方法;5)贝叶斯分位回归联合建模理论及方法;6)部分函数型卷积光滑分位回归模型理论方法;7) 关于混合域数据的广义函数型线性建模;8) 纵向数据复合分位回归理论与应用;9) 相依型函数数据的回归分析研究;10) 半参数非随机缺失分位回归的经验似然推断。以上这些成果具有重要的科学意义和应用前景。

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