  We study here a Gaussian Mixture Model (GMM) with rare events data. In this
case, the commonly used Expectation-Maximization (EM) algorithm exhibits
extremely slow numerical convergence rate. To theoretically understand this
phenomenon, we formulate the numerical convergence problem of the EM algorithm
with rare events data as a problem about a contraction operator. Theoretical
analysis reveals that the spectral radius of the contraction operator in this
case could be arbitrarily close to 1 asymptotically. This theoretical finding
explains the empirical slow numerical convergence of the EM algorithm with rare
events data. To overcome this challenge, a Mixed EM (MEM) algorithm is
developed, which utilizes the information provided by partially labeled data.
As compared with the standard EM algorithm, the key feature of the MEM
algorithm is that it requires additionally labeled data. We find that MEM
algorithm significantly improves the numerical convergence rate as compared
with the standard EM algorithm. The finite sample performance of the proposed
method is illustrated by both simulation studies and a real-world dataset of
Swedish traffic signs.
