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 of the test is also proved. The case of misspecified autocorrelation operator of the ARH(1) process is addressed. The asymptotic equivalence in probability, uniformly in the norm of H, of the empirical processes formulated under known and unknown autocorrelation operator is obtained. 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：本文提出的方法用于检验自回归希尔伯特模型（ARH(1)模型）的拟合优度，该方法基于残差标记的经验过程，是Koul与Stute（1999）所提方法在无限维情形下的推广。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，在原假设下得到了所构造的H值经验过程（同样以H为指标）的渐近性质。极限过程为H值广义（即以H为指标的）维纳过程，从而导出一个渐近分布自由的检验。本文亦证明了该检验的一致性。同时探讨了ARH(1)过程自相关算子设定错误的情形，得到了在已知与未知自相关算子下构造的经验过程在H范数意义下一致的概率渐近等价性。此方法超越了欧几里得框架，使得在流形及球面函数自回归过程的背景下实施拟合优度检验成为可能。