Local Walsh-average-based Estimation and Variable Selection for Spatial Single-index Autoregressive Models

This paper is concerned with spatial single-index autoregressive model (SSIM), where the spatial lag effect enters the model linearly and the relationship between variables is a nonparametric function of a linear combination of multivariate regressors. It addresses challenges related to the curse of dimensionality and interactions among non-independent variables in spatial data. The local Walsh-average regression has proven to be a robust and efficient method for handling single-index models. We extend this approach to the spatial domain, propose a regularized local Walsh-average (RLWA) estimation strategy where the nonparametric component is established by a local Walsh-average approach and the estimation of the parametric part by Walsh-average method. Under specific assumptions, we establish the asymptotic properties of both parametric and nonparametric partial estimators. Additionally, we propose a robust shrinkage method termed regularized local Walsh-average (RLWA) that can construct robust parametric variable selection and robust nonparametric component estimation simultaneously. Theoretical analysis reveals RLWA works beautifully, including consistency in variable selection and oracle property in estimation. We propose a parameter selection process based on a robust BIC-type approach with an oracle property. The effectiveness of the proposed estimation procedure is evaluated through three Monte Carlo simulations and real data applications, demonstrating its performance in finite samples.