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：本文提出的检验希尔伯特自回归模型（ARH(1)模型）拟合优度的方法学，为Koul与Stute（1999）基于残差标记经验过程的方法提供了无限维形式。通过应用希尔伯特值鞅差序列的中心极限定理与泛函中心极限定理，在零假设下得到了所构造的H值经验过程的渐近行为（该过程同样以H为索引）。极限过程为H值广义（即以为H索引）维纳过程，由此可得渐近分布无关检验。同时分析了检验的一致性，并探讨了ARH(1)过程自相关算子设定错误的情形。该方法超越欧几里得框架，可在流形与球面泛函自回归过程中实现拟合优度检验。