MIXTURE-MODELS FOR STUDYING STELLAR POPULATIONS .1. UNIVARIATE MIXTURE-MODELS, PARAMETER-ESTIMATION, AND THE NUMBER OF DISCRETE POPULATION COMPONENTS

Various models have been used to represent the stellar populations of our Galaxy: two-component models (e.g., Populations I and II, disk and halo), three-component models (e.g., thin disk, thick disk, halo), five-component models (e.g., spiral arm, young-, intermediate-, old-disk, and halo), and models in which the number of components is large (a limiting case being Lindblad's continuum model). These models, which are defined by the number of components, the properties of the individual components, and the mixing proportions, provide a framework for explaining the chemical and dynamical history of the Galaxy. The purpose of this paper is (1) to present a unified approach to the analysis of stellar populations through the application of finite mixture models; (2) to examine the statistical properties of univariate finite mixture models; and (3) to review the methods of analysis that are available, in particular, methods for estimating the parameters that describe the underlying stellar populations and for determining the number of discrete stellar populations. For illustrative purposes attention is restricted to the five variables: U, V, W, [Fe/H], and age. Under the assumption that within a stellar population each of these variables follows a Gaussian distribution, and using representative parameter values for solar neighborhood samples, overall mixture distributions, coefficients of skewness and kurtosis, and posterior mixing proportions are computed for two-, three-, and five-component models. Parameter and error estimation are discussed in general terms and are demonstrated in two simulation experiments designed to assess the detectability of a thick disk in samples of solar neighborhood stars. In addition, the [Fe/H] distribution of 120 Galactic globular clusters is analyzed. Methods for determining the number of discrete population components are discussed briefly.