标题： 自适应弹性网络估计用于稀疏扩散过程 显示英文标题

标题： Adaptive Elastic-Net estimation for sparse diffusion processes

摘要： 惩罚估计方法近年来因其在处理高维随机系统方面的有效性而受到广泛关注。 本文中，我们提出了一个针对平稳扩散过程的自适应弹性网络估计器，该过程在高频采样方案下被观测到。 我们的方法结合了拟似然函数的最小二乘近似以及自适应的$\ell_1$和$\ell_2$正则化。 这种方法在提高预测准确性与可解释性的同时，能够有效地恢复模型稀疏的潜在结构。 基于分析高维场景的精神，通过推导$\ell_2$估计误差的高概率非渐近界，我们提供了（块对角线）估计器性能的有限样本保证。 这些结果补充了在高频渐近框架下已确立的oracle性质，并具有混合收敛速度，确保相关交互作用的一致选择并达到最优收敛率。 此外，我们利用这些结果来分析一步预测，提供$\ell_1$预测误差的非渐近控制。 通过模拟和真实数据应用评估了我们的方法的表现，展示了其在变量高度相关的场景中的有效性。 显示英文摘要

摘要： Penalized estimation methods for diffusion processes and dependent data have recently gained significant attention due to their effectiveness in handling high-dimensional stochastic systems. In this work, we introduce an adaptive Elastic-Net estimator for ergodic diffusion processes observed under high-frequency sampling schemes. Our method combines the least squares approximation of the quasi-likelihood with adaptive $\ell_1$ and $\ell_2$ regularization. This approach allows to enhance prediction accuracy and interpretability while effectively recovering the sparse underlying structure of the model. In the spirit of analyzing high-dimensional scenarios, we provide finite-sample guarantees for the (block-diagonal) estimator's performance by deriving high-probability non-asymptotic bounds for the $\ell_2$ estimation error. These results complement the established oracle properties in the high-frequency asymptotic regime with mixed convergence rates, ensuring consistent selection of the relevant interactions and achieving optimal rates of convergence. Furthermore, we utilize our results to analyze one-step-ahead predictions, offering non-asymptotic control over the $\ell_1$ prediction error. The performance of our method is evaluated through simulations and real data applications, demonstrating its effectiveness, particularly in scenarios with strongly correlated variables.