Estimation beyond Missing (Completely) at Random

We study the effects of missingness on the estimation of population parameters. Moving beyond restrictive missing completely at random (MCAR) assumptions, we first formulate a missing data analogue of Huber's arbitrary $\epsilon$-contamination model. For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption, and a robust error term, depending on $\epsilon$, that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable $\epsilon$-contamination models, where an MCAR version of a base distribution $P$ is contaminated by an arbitrary missing not at random (MNAR) version of $P$. These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable $\epsilon$-contamination classes is possible even when $\epsilon$ and the proportion of missingness converge (slowly) to 1. We extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response, and also demonstrate that our methods can be made adaptive to the case of unknown $\epsilon$.