The effect of sample size and missingness on inference with missing data

When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This article 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.