In small area estimation, it is a smart strategy to rely on data measured over time. However, linear mixed models struggle to properly capture time dependencies when the number of lags is large. Given the lack of published studies addressing robust prediction in small areas using time-dependent data, this research seeks to extend M-quantile models to this field. Indeed, our methodology successfully addresses this challenge and offers flexibility to the widely imposed assumption of unit-level independence. Under the new model, robust bias-corrected predictors for small area linear indicators are derived. Additionally, the optimal selection of the robustness parameter for bias correction is explored, contributing theoretically to the field and enhancing outlier detection. For the estimation of the mean squared error (MSE), a first-order approximation and analytical estimators are obtained under general conditions. Several simulation experiments are conducted to evaluate the performance of the fitting algorithm, the new predictors, and the resulting MSE estimators, as well as the optimal selection of the robustness parameter. Finally, an application to the Spanish Living Conditions Survey data illustrates the usefulness of the proposed predictors.

翻译：在小区域估计中，依赖时间序列数据是一种明智的策略。然而，当滞后阶数较大时，线性混合模型难以有效捕捉时间依赖性。鉴于目前缺乏利用时间依赖数据进行小区域稳健预测的公开研究，本研究旨在将M-分位数模型扩展至该领域。事实上，我们的方法成功解决了这一挑战，并对广泛采用的单元级独立性假设提供了灵活性。基于新模型，我们推导了小区域线性指标的稳健偏差校正预测量。此外，本研究探索了偏差校正中稳健参数的最优选择，为该领域提供了理论贡献并增强了异常值检测能力。针对均方误差的估计，我们在一般条件下获得了一阶近似解与解析估计量。通过多组模拟实验，评估了拟合算法、新预测量及其对应均方误差估计量的性能，以及稳健参数的最优选择效果。最后，基于西班牙生活条件调查数据的应用实例验证了所提预测方法的实用价值。