  The ubiquity of missing values in real-world datasets poses a challenge for
statistical inference and can prevent similar datasets from being analyzed in
the same study, precluding many existing datasets from being used for new
analyses. While an extensive collection of packages and algorithms have been
developed for data imputation, the overwhelming majority perform poorly if
there are many missing values and low sample sizes, which are unfortunately
common characteristics in empirical data. Such low-accuracy estimations
adversely affect the performance of downstream statistical models. We develop a
statistical inference framework for regression and classification in the
presence of missing data without imputation. Our framework, RIFLE (Robust
InFerence via Low-order moment Estimations), estimates low-order moments of the
underlying data distribution with corresponding confidence intervals to learn a
distributionally robust model. We specialize our framework to linear regression
and normal discriminant analysis, and we provide convergence and performance
guarantees. This framework can also be adapted to impute missing data. In
numerical experiments, we compare RIFLE to several state-of-the-art approaches
(including MICE, Amelia, MissForest, KNN-imputer, MIDA, and Mean Imputer) for
imputation and inference in the presence of missing values. Our experiments
demonstrate that RIFLE outperforms other benchmark algorithms when the
percentage of missing values is high and/or when the number of data points is
relatively small. RIFLE is publicly available at
https://github.com/optimization-for-data-driven-science/RIFLE.
