Bumps, Mixtures and Models with Measurement Errors

Most astronomical data are pre-processed data, come with information on their measurement errors. To make optimal use of these precious data, it is important not to ignore the measurement errors in our statistical analyses. In this talk, we first illustrate why ``how do galaxies form" -- one of the most important questions of modern astronomy -- is related to bump hunting and mixture model fitting. We then offer solutions that incorporate measurement errors in the study of bumps and mixtures. The errors can be homogeneous or nonhomogeneous; their distribution affects the convergence rates of our procedures (parametric or nonparametric). It is shown that naive procedures, ignoring the measurement errors, can miss important information of a data set, or is inconsistent to the "parameter" of interest. Finally, we analyze halo data, obtained recently from the telescope, using our new procedures. The data show some evidence of a ``bottom-up'' galaxy formation process.