Functional data analysis (FDA) is a branch of statistics that analyzes data providing information about curves, surfaces or anything else varying over a continuum. In its most general form, under an FDA framework, each sample element of functional data is considered to be a random function.

Popularized by Ramsay and Silverman [1,2], statistics for functional data analysis have attracted considerable research interest because of its wide applications in many practical fields, such as medicine, economics and linguistics. For an introduction to the topics, we can refer to the monographs of Ramsay and Silverman [3] for parametric models, and Ferraty and Vieu [4] for nonparametric models.

In this paper, the following functional non-parametric regression model is considered. whereY is a scalar response variable,$\chi$ is a covariate taking value in a subset$S_{\mathfrak{F}}$ of an infinite-dimensional functional space$\mathfrak{F}$ endowed with a semi-metric$d(·,·)$.$m(·)$ is the unknown regression operator from$S_{\mathfrak{F}}$ to$\mathfrak{R}$, and the random error$ϵ$ satisfies$E\left(ϵ\right|\chi )=0,a.s.$

For the estimation of model (1), Ferraty and Vieu [5] investigated the classical functional Nadaraya-Watson (N-W) kernel type estimator of$m(·)$ and obtained the asymptotic properties with rates in the case of$\alpha$-mixing functional data. Ling and Wu [6] studied the modified N-W kernel estimate and derived the asymptotic distribution for strong mixing functional time series data, Baíllo and Grane [7] proposed a functional local linear estimate based on the local linear idea. In this paper, we focus on the k-nearest neighbors (kNN) method for regression model (1). The kNN method, as one of the most simple and traditional nonparametric techniques, is often used as a nonparametric classification method. The kNN method was first developed by Evelyn Fix and Joseph Hodges in 1951 [8] and then expanded by Thomas Cover [9]. In our kNN regression, the input consists of the k-closest training examples in a dataset, whereas the output is the property value for the object. This value is the average of the values of the k-nearest neighbors. Under independent samples, research in kNN regression mostly focuses on the estimation of the continuous regression function$m\left(\chi \right)$. For example, Burba et al. [10] investigated the kNN estimator based on the idea of the local adaptive bandwidth of functional explanatory variables. The papers [11,12,13,14,15,16,17,18], and others, obtained the asymptotic behavior of nonparametric regression estimators for functional data in independent and dependent cases. Further, Kudraszow and Vieu [19] obtained asymptotic results for a kNN generalized regression estimator when the observed variables take values in an abstract space. Kara-Zaitri et al. [20] provided an asymptotic theory for several different target operators and some simulated experiences, including regression, conditional density, conditional distribution and hazard operators. However, functional observations often behave with correlation, including satisfying some form of negative dependence or negative association.

Negatively associated (NA) sequences were introduced by Joag-Dev and Proschan in [21]. Random variables${\left\{Y_i\right\}}_{1\le i\le n}$ are said to be NA, if for every pair of disjoint subsets$A,B\subset \{1,2,\dots ,n\},$ or equivalently, wheref andg are coordinatewise non-decreasing, such that this covariance exists. An infinite sequence${\left\{Y_n\right\}}_{n\ge 1}$ is NA if every finite subcollection is NA.

For example, if${\left\{Y_i\right\}}_{1\le i\le n}$ follows permutation distributions, where$\{Y_1,Y_2,\dots ,Y_n\}=\{y_1,y_2,\dots ,y_n\}$ always and$y_1\le y_2\le \dots \le y_n$ aren real numbers, then${\left\{Y_i\right\}}_{1\le i\le n}$ is NA.

Whereas kNN regression under NA sequences has not been explored in the literature, in this paper, we extend the kNN estimation of functional data from the case of independent samples to NA sequences.

Let a pair${\left\{({\chi }_i,Y_i)\right\}}_{i=1,\dots ,n}$ be a sample of NA pairs in$(\chi ,Y)$, which is a random vector valued in the$\mathfrak{F}\times \mathfrak{R}$.$(\mathfrak{F},d)$ is a semi-metric space,$\mathfrak{F}$ is not necessarily of the finite dimension and we do not suppose the existence of a density for the functional random variable$\chi$. For a fixed$\chi \in \mathfrak{F}$, the closed ball with$\chi$ as the center and$ϵ$ as the radius is denoted as:

The kNN regression estimator [10] is defined as follows: where$K(·)$ is the kernel function supported on$[0,\infty )$.$H_{n,k}\left(\chi \right)$ is a positive random variable that depends on$({\chi }_1,{\chi }_2,\dots ,{\chi }_n)$ and is defined by: obviously, the kNN estimator can be seen as an expansion to a random locally adaptive neighborhood of the traditional kernel method [5] defined as: where$h_n\left(\chi \right)$ is a sequence of positive real numbers such as$h_n\left(\chi \right)\to 0$ a.s.$n\to \infty$.

This paper is organized as follows. The main results of our paper about the asymptotic behavior of the kNN estimators using a data-driven random number of neighbors are given inSection 2.Section 3 illustrates the numerical performance of the proposed method, including nonparametric functional regression analysis of the sea level surface temperature (SST) data for the El Ni$\tilde{n}$o area (0–100 S, 800–900 W). The technical proofs are postponed toSection 4. Finally,Section 5 is devoted to comments on the results and to related perspectives for the future.

In this section, we focus on the asymptotic property of the kNN regression estimator and need to state the convergence rate of an estimator.

One says that the rate of almost complete convergence of a sequence$\{Y_n,n\ge 1\}$ toY is of order$u_n$ if only if for any$ϵ>0$, and we write$Y_n-Y=O_{a.co.}\left(u_n\right)$(see for instance [5]). By the Borel-Cantelli lemma, this implies that$\frac{Y_n-Y}{u_n}\to 0$ almost surely, so almost complete convergence is a stronger result than almost sure convergence.

Our results are stated under some mild assumptions we gather below for easy references. Throughout the paper, we will denote by$C,C_1,C^{\prime }$ some positive generic constants, which may be different in various places.

In order to prove the main results, we give some lemmas. Let${(A_i,B_i)}_{i=1,2,\dots ,n}$ ben random pairs valued in$(\mathsf{\Omega }\times \mathfrak{R},\mathfrak{A}\times \mathfrak{B}(\mathfrak{R}\left)\right)$, where$(\mathsf{\Omega },\mathfrak{A})$ is a general measurable space. Let$S_{\mathsf{\Omega }}$ be a fixed subset of$\mathsf{\Omega }$,$G(·,(\chi ,·)):\mathfrak{R}\times (S_{\mathsf{\Omega }}\times \mathsf{\Omega })\to {\mathfrak{R}}^{+}$ be a measurable function, for$\forall t,t^{\prime }\in \mathfrak{R}$,${D_n\left(\chi \right)}_{n\in \mathbb{N}}$ is a sequence of random real variables (r.r.v.), and$c(·):S_{\mathsf{\Omega }}\to \mathfrak{R}$ is a nonrandom function such that${sup}_{\chi \in S_{\mathsf{\Omega }}}\left|c\left(\chi \right)\right|<\infty$. For$\forall \chi \in S_{\mathsf{\Omega }},n\in \mathbb{N}/\left\{0\right\}$, we define:

The proof of Lemma 1 is not presented here because it follows, step by step, the same argument in Burba et al. [10], Kudraszow and Vieu [19].