Automatic and location-adaptive estimation in functional single-index regression

Abstract

This paper develops a new automatic and location-adaptive procedure for estimating regression in a Functional Single-Index Model (FSIM). This procedure is based on $k$-Nearest Neighbours ($k$NN) ideas. The asymptotic study includes results for automatically data-driven selected number of neighbours, making the procedure directly usable in practice. The local feature of the $k$NN approach insures higher predictive power compared with usual kernel estimates, as illustrated in some finite sample analysis. As by-product we state as preliminary tools some new uniform asymptotic results for kernel estimates in the FSIM model.

Figures (5)

• Figure 1: Sample of 50 curves $X$ (left panel) together with the corresponding scatter plot of $\{\left(\left<\theta_0,X\right>, Y\right)\}$ (right panel).
• Figure 2: Average of the cross-validation functions obtained from both the kernel-based estimators (left panel) and the $k$NN-based ones as function of the bandwidth ($h$) and the number of neighbours ($k$), respectively. The dashed lines show the average of the cross-validation functions when optimal values for $h$ (left panel) and $k$ (right panel) are considered. From top to bottom, the pairs (solid curve, dashed line) correspond to $n=50,100,200$.
• Figure 3: Average of the MSEP functions obtained from both the kernel-based estimators (left panel) and the $k$NN-based ones as function of the bandwidth ($h$) and the number of neighbours ($k$), respectively. The dashed lines show the average of the MSEP functions when values for $h$ (left panel) and $k$ (right panel) obtained from the cross-validation method are considered. From top to bottom, the pairs (solid curve, dashed line) correspond to $n=50,100,200$.
• Figure 4: Sample of 100 absorbance curves $X$ (left panel) together with their second derivatives $X^{(2)}$ (right panel).
• Figure 5: Left panel: Estimate of the functional direction $\theta_0$. Right panel: estimates of the regression $r(\cdot)$ by means of the $k$NN-based (solid line) and kernel-based (dashed line) estimates.

Theorems & Definitions (15)

• Proposition 3.1
• Remark 3.2
• Theorem 3.3
• Corollary 3.4
• Remark 3.5
• Corollary 4.1
• Remark 4.2
• Lemma A.1: Theorem 2.14.1 in *van1196, p. 239
• Lemma A.2: Theorem 3.1 in *dony, p. 314
• Lemma A.3: Bernstein type inequality in *dony, p. 321