  When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist)
obtained from partial data valid? This paper answers this question by offering
a new asymptotic theory about inference with missing data that is more general
than existing theories. It proves that as the sample size increases and the
extent of missingness decreases, the average-loglikelihood function generated
by partial data and that ignores the missingness mechanism will converge in
probability to that which would have been generated by complete data; and if
the data are Missing at Random, this convergence depends only on sample size.
Thus, inferences from partial data, such as posterior modes, confidence
intervals, likelihood ratios, test statistics, and indeed, all quantities or
features derived from the partial-data loglikelihood function, will be
consistently estimated. Additionally, the missing data mechanism has
asymptotically no effect on parameter estimation and hypothesis testing if the
data are Missing at Random. This adds to previous research which has only
proved the consistency and asymptotic normality of the posterior mode.
Practical implications are discussed, and the theory is illustrated through
simulation using a previous study of International Human Rights Law.
