  This paper studies hypothesis testing and parameter estimation in the context
of the divide and conquer algorithm. In a unified likelihood based framework,
we propose new test statistics and point estimators obtained by aggregating
various statistics from $k$ subsamples of size $n/k$, where $n$ is the sample
size. In both low dimensional and high dimensional settings, we address the
important question of how to choose $k$ as $n$ grows large, providing a
theoretical upper bound on $k$ such that the information loss due to the divide
and conquer algorithm is negligible. In other words, the resulting estimators
have the same inferential efficiencies and estimation rates as a practically
infeasible oracle with access to the full sample. Thorough numerical results
are provided to back up the theory.
