  We study the statistical properties of an estimator derived by applying a
gradient ascent method with multiple initializations to a multi-modal
likelihood function. We derive the population quantity that is the target of
this estimator and study the properties of confidence intervals (CIs)
constructed from asymptotic normality and the bootstrap approach. In
particular, we analyze the coverage deficiency due to finite number of random
initializations. We also investigate the CIs by inverting the likelihood ratio
test, the score test, and the Wald test, and we show that the resulting CIs may
be very different. We propose a two-sample test procedure even when the MLE is
intractable. In addition, we analyze the performance of the EM algorithm under
random initializations and derive the coverage of a CI with a finite number of
initializations.
