  We present a complete framework for determining the asymptotic (or
logarithmic) efficiency of estimators of large deviation probabilities and rate
functions based on importance sampling. The framework relies on the idea that
importance sampling in that context is fully characterized by the joint large
deviations of two random variables: the observable defining the large deviation
probability of interest and the likelihood factor (or Radon-Nikodym derivative)
connecting the original process and the modified process used in importance
sampling. We recover with this framework known results about the asymptotic
efficiency of the exponential tilting and obtain new necessary and sufficient
conditions for a general change of process to be asymptotically efficient. This
allows us to construct new examples of efficient estimators for sample means of
random variables that do not have the exponential tilting form. Other examples
involving Markov chains and diffusions are presented to illustrate our results.
