稀疏纵向函数型数据均值的优化估计与拟合优度检验

中文摘要 AI 辅助

我们研究纵向函数型数据的均值函数，其中每个受试者在一般域上于随机访问时间贡献少量完整轮廓。均值在时间方向上投影到标准正交基上，每个系数函数通过轮廓的加权平均来估计。我们考虑确定性和随机权重，包括完全不需要设计密度的闭式数据驱动权重。对于每种加权方案，我们推导了积分二次风险的非渐近界和显式的最优截断水平。所有方案共享相同的收敛速率，我们证明该速率在相应的系数衰减类上是极小极大最优的，仅领先常数不同。我们还提供了均值函数的拟合优度检验，并在原假设和备择假设下推导了其分布的高斯近似的非渐近界。我们的估计和检验程序易于实现、快速，并且在应用中表现良好。

英文摘要

We study the mean function of longitudinal functional data, where each subject contributes a small number of complete profiles over a general domain, observed at random visit times. The mean is projected onto an orthonormal basis in the time direction, and each coefficient function is estimated by a weighted average of the profiles. We consider deterministic and random weights, including closed-form, data-driven weights that dispense with the design density entirely. For each weighting scheme we derive non-asymptotic bounds on the integrated quadratic risk and an explicit optimal truncation level. All schemes share the same convergence rate, which we show to be minimax optimal over the corresponding coefficient-decay class, only the leading constants differ. We also provide a goodness-of-fit test for the mean function, and derive non-asymptotic bounds for the Gaussian approximation of its distribution under both the null and alternative hypotheses. Our estimation and testing procedures are easy to implement, fast, and perform well in applications.

发表机构

• Univ. Rennes, Ensai, CREST-UMR 9194（雷恩大学，ENSAI，CREST）
• Ensai

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