We consider the estimation of the parameters $s = (\nu, \alpha_1, \alpha_2, \cdots, \alpha_T)$ of a cumulative INAR($\infty$) process based on finite observations under the assumption $\sum_{k=1}^T \alpha_k < 1$ and $\sum_{k=1}^T\alpha_k^2<\frac12$. The parameter space is modeled as a Euclidean space $\mathfrak{l}^2$, with an inner product defined for pairs of parameter vectors. The primary goal is to estimate the intensity function $\Phi_s(t)$, which represents the expected value of the process at time $t$. We introduce a Least-Squares Contrast $\gamma_T(f)$, which measures the distance between the intensity function $\Phi_f(t)$ and the true intensity $\Phi_s(t)$. We further show that the contrast function $\gamma_T(f)$ can be used to estimate the parameters effectively, with an associated metric derived from a quadratic form. The analysis involves deriving upper and lower bounds for the expected values of the process and its square, leading to conditions under which the estimators are consistent. We also provide a bound on the variance of the estimators to ensure their asymptotic reliability.

翻译：我们考虑基于有限观测数据估计累积INAR($\infty$)过程的参数 $s = (\nu, \alpha_1, \alpha_2, \cdots, \alpha_T)$，假设满足 $\sum_{k=1}^T \alpha_k < 1$ 且 $\sum_{k=1}^T\alpha_k^2<\frac12$。参数空间被建模为欧几里得空间 $\mathfrak{l}^2$，其中定义了参数向量对的内积。主要目标是估计强度函数 $\Phi_s(t)$，该函数表示过程在时刻 $t$ 的期望值。我们引入了最小二乘对比函数 $\gamma_T(f)$，用于度量强度函数 $\Phi_f(t)$ 与真实强度 $\Phi_s(t)$ 之间的距离。进一步证明，对比函数 $\gamma_T(f)$ 可有效用于参数估计，其相关度量源自二次型。分析涉及推导过程及其平方的期望值的上下界，从而得出估计量具有一致性的条件。我们还给出了估计量方差的界，以确保其渐近可靠性。