We propose a nonlinear function-on-function regression model where both the covariate and the response are random functions. The nonlinear regression is carried out in two steps: we first construct Hilbert spaces to accommodate the functional covariate and the functional response, and then build a second-layer Hilbert space for the covariate to capture nonlinearity. The second-layer space is assumed to be a reproducing kernel Hilbert space, which is generated by a positive definite kernel determined by the inner product of the first-layer Hilbert space for $X$--this structure is known as the nested Hilbert spaces. We develop estimation procedures to implement the proposed method, which allows the functional data to be observed at different time points for different subjects. Furthermore, we establish the convergence rate of our estimator as well as the weak convergence of the predicted response in the Hilbert space. Numerical studies including both simulations and a data application are conducted to investigate the performance of our estimator in finite sample.

翻译：我们建议一个非线性函数在功能回归模型, 共变和响应都是随机的功能。 非线性回归分两个步骤进行: 我们先建造希尔伯特空间, 以适应功能共变和功能反应, 然后为共变建立二级希尔伯特空间, 以捕捉非线性。 第二层空间被假定为复制内核希尔伯特空间, 这个空间是由第一层希尔伯特空间的内产物确定的一个肯定的内核产生的, 这个结构被称为嵌套的希尔伯特空间。 我们开发了估算程序, 以实施拟议方法, 允许在不同时间点观测不同主题的功能数据。 此外, 我们建立了我们的天主的趋同率, 以及希伯特空间预测反应的薄弱趋同性。 包括模拟和数据应用在内的内核研究正在进行, 以调查我们在定点样本中的估量器的性能。