Multivariate Estimation for Astronomy

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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. ***