Shape‐restricted statistical inference for non‐ignorable missing data under a general additive model

Although ubiquitous in many areas, missing data problems become challenging when the missingness of the outcome depends on itself, which means the data are non‐ignorable missing. To alleviate the risk of model misspecification and balance interpretation and efficiency, we model the non‐missingness probability by a logistic model with a general additive covariate effect under shape restrictions. Each additive component is assumed to satisfy certain shape restrictions, such as monotone increasing/decreasing, convexity/concavity, or a combination of these. We develop a shape‐restricted and tuning‐parameter‐free estimator for the population outcome mean with the help of an instrument variable. We systematically establish the consistency, convergence rates, and asymptotic normalities of the proposed estimators. Our numerical results indicate that the proposed shape‐restricted estimator has comparable performance to competing estimators with parametric models when the parametric models are correct, and outperforms them when the parametric models are misspecified. Finally, our method is applied to two real datasets providing more interpretable results than its competitors.