  (abridged) We develop an algorithm for estimating parameters of a
distribution sampled with contamination, employing a statistical technique
known as ``expectation maximization'' (EM). Given models for both member and
contaminant populations, the EM algorithm iteratively evaluates the membership
probability of each discrete data point, then uses those probabilities to
update parameter estimates for member and contaminant distributions. The EM
approach has wide applicability to the analysis of astronomical data. Here we
tailor an EM algorithm to operate on spectroscopic samples obtained with the
Michigan-MIKE Fiber System (MMFS) as part of our Magellan survey of stellar
radial velocities in nearby dwarf spheroidal (dSph) galaxies. These samples are
presented in a companion paper and contain discrete measurements of
line-of-sight velocity, projected position, and Mg index for ~1000 - 2500 stars
per dSph, including some fraction of contamination by foreground Milky Way
stars. The EM algorithm quantifies both dSph and contaminant distributions,
returning maximum-likelihood estimates of the means and variances, as well as
the probability that each star is a dSph member. Applied to our MMFS data, the
EM algorithm identifies more than 5000 probable dSph members. We test the
performance of the EM algorithm on simulated data sets that represent a range
of sample size, level of contamination, and amount of overlap between dSph and
contaminant velocity distributions. The simulations establish that for samples
ranging from large (N ~3000) to small (N~30), the EM algorithm distinguishes
members from contaminants and returns accurate parameter estimates much more
reliably than conventional methods of contaminant removal (e.g., sigma
clipping).
