Multivariate Estimation for Astronomy

天文研究中的许多重要问题需要的模型往往是复合的和非线性的，比标准多变量分析中假设的模型要复杂得多。虽然这些模型有时可以进行分析计算，但它们通常只能通过蒙特卡罗模拟来表示，特别是在数据收集中涉及选择偏差时。在这种情况下，经验分布不接近底层总体分布，标准参数估计不适用。研究人员研究了一种基于多元数据集在一维空间上的投影的估计程序，以导出隐藏参数的最优值和约束，基于投影的kolmogorov - smirnov型统计将用于将数据与模型进行比较。新统计量的渐近一致性和渐近分布将使用经验过程的近似理论进行评估。当应用于天文问题时，即使对于有偏差的多变量数据和复杂模型，也可以计算出“最佳拟合”模型参数。天文学研究中的许多重要问题涉及将复杂的天体物理模型应用于具有许多变量的数据集：大爆炸以来星系的演化，银河系中物质的分布，球状星团中最古老恒星的年龄。%%% %天文学家寻求深入了解模型的有效性以及模型参数与数据一致的范围。这些模型通常非常复杂，而且真实的数据集经常受到已知选择偏差的影响（例如，只检测到较亮的星系）。我们开发和应用必要的数学和统计工具来处理这些问题。该项目是促进两门学科智力整合的长期努力的一部分：统计学，它拥有理解数据的复杂工具；和天文学，它面对的是关于我们的物理宇宙的基本问题。＊＊＊

DMS 9626189 Babu Many important problems in astronomical research require models that are often composite and nonlinear, far more complex than the models assumed in standard multivariate analysis. While these models can sometimes be calculated analytically, they often can only be represented by Monte Carlo simulations, particularly when selection biases are involved in the data collection. In such cases, the empirical distribution does not approach the underlying population distribution and standard parameter estimation is inapplicable. The investigators study an estimation procedure, based on projections of multivariate datasets on to 1-dimensional spaces, to derive optimal values and constraints on the hidden parameters Kolmogorov-Smirnov-type statistics based on the projections will be used to compare the data with models. Asymptotic consistency and the asymptotic distribution of the new statistics will be evaluated using approximation theory of empirical processes. When applied to astronomical problems, `best-fit' model parameters may be computed even for biased multivariate data and complicated models. %%% Many important problems in astronomical research involve applying complicated astrophysical models to datasets with many variables: the evolution of galaxies since the Big Bang, the distribution of matter in our Galaxy, the age of the oldest stars in globular clusters. The astronomer seeks insight into the validity of the models and the range of model parameters consistent with the data. The models are often very complex, and real datasets frequently suffer from known selection biases (e.g. only the brighter galaxies are detected). We develop and apply the mathematical and statistical tools necessary to treat such problems. This project is part of a long-term effort to promote intellectual integration of two disciplines: statistics, which has sophisticated tools for understanding data; and astronomy, which confronts fundamental questions a bout our physical Universe. ***