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：本文提出的自回归希尔伯特模型（ARH(1)模型）拟合优度检验方法，基于Koul与Stute（1999）提出的残差标记经验过程框架，给出了无限维形式表述。通过应用希尔伯特值鞅差序列的中心极限定理与函数中心极限定理，在原假设下推导了所构造的H值经验过程的渐近行为，该过程以H为指标集。极限过程为H值广义（即以H为指标集）维纳过程，从而得到了渐近分布自由的检验，并证明了检验的一致性。针对ARH(1)过程自相关算子设定错误的情形，论证了基于已知与未知自相关算子构造的两个经验过程在H范数意义下依概率渐近等价。该方法突破了欧几里得空间限制，可在流形及球面函数自回归过程中实现拟合优度检验。