The presented methodology for testing the goodness-of-fit of an Autoregressive Hilbertian model (ARH(1) model) provides an infinite-dimensional formulation of the approach proposed in Koul and Stute (1999), based on empirical process marked by residuals. Applying a central and functional central limit result for Hilbert-valued martingale difference sequences, the asymptotic behavior of the formulated H-valued empirical process, also indexed by H, is obtained under the null hypothesis. The limiting process is H-valued generalized (i.e., indexed by H) Wiener process, leading to an asymptotically distribution free test. Consistency is also analyzed. The case of misspecified autocorrelation operator of the ARH(1) process is addressed as well. Beyond the Euclidean setting, this approach allows to implement goodness of fit testing in the context of manifold and spherical functional autoregressive processes.

翻译：本文提出的检验自回归Hilbert模型（ARH(1)模型）拟合优度的方法，将Koul和Stute（1999）基于残差标记经验过程的检验方法推广至无穷维框架。利用Hilbert值鞅差序列的中心极限定理与泛函中心极限定理，在原假设下得到了所构造的H值经验过程（亦以H为指标集）的渐近性质。该极限过程为H值广义（即以H为指标集）维纳过程，从而给出渐近分布自由的检验。本文还分析了检验的一致性，并讨论了ARH(1)过程自相关算子设定错误的情形。超越欧几里得设定，该方法可用于流形与球面泛函自回归过程的拟合优度检验。