The classical $k$-means clustering, based on distances computed from all data features, cannot be directly applied to incomplete data with missing values. A natural extension of $k$-means to missing data is to involve only the observed positions in clustering, which is equivalent to imputing missing values by corresponding cluster means. However, for data missing not at random (MNAR), since missingness is related to data values, such a mean-imputation-based method may lead to the distortion of estimated cluster centers, resulting in a poor clustering result. Since MNAR mechanisms are very common in reality, it is necessary to improve the performance of $k$-means-based clustering methods for such data. In this paper, we focus on a magnitude-decaying MNAR scenario where data is more likely to be missing at positions with smaller absolute values, and we propose a novel $k$-means clustering method based on the constraint of the size of imputation values, which enjoys a good mathematical interpretation. Moreover, we establish the statistical consistency of the estimated cluster centers of the proposed method to the true cluster centers of fully observed data, and solve the optimization of the proposed loss function via an alternative minimization algorithm. Simulation experiments verify the effect of the proposed method in improving clustering results and reducing the bias of estimated cluster centers. Applications to real-world missing data further show the utility of the proposed method.
In offline Reinforcement Learning, immediate rewards in logged batch data are often unobserved due to sparse or irregular record-keeping, or censored beyond certain reward values. This issue arises in practical settings, including health care and marketing. We investigate off-policy evaluation (OPE) in finite-horizon Markov decision processes when rewards are missing not at random (MNAR), which breaks ignorability and induces selection bias even after conditioning on states and actions. To address this, we formalize a reward-dependent propensity model and use future states as shadow variables to identify the full-data conditional mean reward. We further introduce a bridge function that recovers the conditional mean reward without explicitly modeling the MNAR mechanism, and estimate it via a min-max procedure to avoid double sampling. Building upon these identification results, we propose an Fitted-Q-Evaluation-style estimator that propagates the recovered rewards while allowing target policies to depend on past missingness indicators. Finally, we establish consistency and finite-sample error bounds for our OPE estimator, and show through experiments the strong performance of our method compared to existing methods on simulated and MIMIC-III Sepsis data.