Adaptive Mode Learning for Nonstationary Streaming Data

We develop a novel framework for adaptive conditional mode estimation in streaming environments with evolving, nonstationary data generating processes. In contrast to traditional mean or quantile estimators, which can be suboptimal under skewed or multimodal distributions, our approach targets the conditional mode to provide a robust and interpretable summary of the most likely outcome given covariates. The proposed estimator relies on a recursive kernel update of the joint density using exponentially weighted observations, enabling real-time computation with bounded memory. To accommodate temporal distributional drift, we introduce a prediction-discrepancy-driven bandwidth adaptation mechanism and a cross-validated forgetting strategy that tune parameters online. We develop theoretical results under both stationary and nonstationary settings, including pointwise and uniform consistency results, non-asymptotic error bounds, and minimax-optimal convergence rate over Hölder classes. From an online learning perspective, we derive a sublinear cumulative regret bound and confirm the estimator’s optimality through a matching minimax lower bound. The method is further shown to be robust to heavy-tailed noise and locally adaptive to unknown smoothness. Simulation studies and an empirical application to bike-sharing demand data demonstrate the estimator’s practical adaptability and predictive accuracy.