Approximately calibrated small sample inference about means from bivariate normal data with missing values

Frequency properties of approximate Bayesian posterior probability intervals are considered for a small bivariate sample with data missing on one variable. For a class of priors, approximations to the posterior distribution based on matching moments of the t distribution are derived, and compared with the true distributions computed numerically. Coverage properties of highest posterior density intervals for three choices of prior are evaluated by simulation, and compared with other solutions. The simulations suggest that a second moment t approximation combined with the Jeffreys' prior for the bivariate distribution provides intervals that are quite well calibrated, in the sense of having approximate or slightly conservative coverage for a wide range of values of the underlying parameters. The use of calibration to select a suitable reference prior seems to have potential for a large number of problems.