Making up for missing data by maximum likelihood estimation: an application to Bernoulli variables

Statistical parameter estimation from incomplete or lacunar data series is an oftencountered issue in real settings, an issue for which the user has at his disposal a handful of solutions, from simple and multiple imputation to substitution of an average value, winsorization, and notably least squares estimation (LSQ, or MC in the article) and maximum likelihood estimation (ML, or MV in the article). LSQ and ML allow to fill in a gap in the series by an enlightened and precise estimation of the missing information, a feat that none of the other methods approaches. This advantage of LSQ and ML over the other less appropriate and precise methods is tied up with their drawback: one must know explicitly the probability function of the variable at stake. LSQ and ML estimates are frequently but not always equal and, if LSQ estimation is best suited to and well documented for real variable distributions, it is much less suitable for integer variables. The present article explores ML estimation under conditions of missing data for a few instances of integer Bernoulli variables, namely the Binomial, Geometric, Pascal (or Negative Binomial), Constrained Pascal, and Poisson distributions. Examples with calculations and tables are provided.