# Method of Sieves with Data Driven Basis Functions

Consider a nonparametric regression problem with i.i.d. sampled data $$(y_1,x_1), (y_2,x_2),\ldots, (y_n,x_n)$$ and regression function $$y_i = g_0(x_i) + \varepsilon_i,\quad \mathbb E[\varepsilon_i | x_i] = 0$$ One common approach to fitting $$g_0$$ non-parametrically is the so-called method of Sieves where for each $$K$$, we use some basis functions $$p^K(x) = (p^{1K}(x),\ldots,p^{KK}(x))$$ and we increase the complexity of the model $$K$$ as a function of $$n$$ so that $$\hat g(x) = (p^{K(n)})'\hat\beta_{K(n)}$$ where $$\hat\beta_{K(n)} = \underset{\beta}{\mathrm{argmin}} \sum_{i=1}^n (y_i - (p^K(x_i))'\beta)^2$$.

I am in particular interested in a setting like Newey (1997), where we are interested in estimating $$g_0$$ for the purposes of computing some functional $$\theta_0 = a(g_0)$$ via a plug-in estimator $$\hat \theta_n = a(\hat g_n)$$. The paper is able to show asymptotic normality of such an estimator under certain conditions. In the paper, it is essentially assumed that the $$p^K$$ are fixed ex-ante. I am curious about a case where the $$p^K$$ are instead formed in a data-driven way and hence are allowed to be stochastic. It seems to me that the proof given in the paper could be modified to accommodate for $$p^K$$ being data driven, but I was wondering if there is already literature on a setting like this.

## 1 Answer

I have been thinking about similar questions in the functional regression context. Data driven orthogonal basis selection for functional data analysis is a paper I have found, though it's not for the setup in your question. (I wanted to comment but don't have enough reputation).

https://doi.org/10.1016/j.jmva.2021.104868

• Hi, welcome to CV. Please add the reference of the paper you cite in case your link dies in the future. Thanks Commented Dec 4, 2022 at 21:16