We in this paper propose a robust estimation technique for a first-order autoregressive process characterized by the root \(\rho _n=1+c/k_n\) , employing a kernel mode-based objective function construed on the mode value. Compared to traditional least squares or maximum likelihood approaches, the newly developed method exhibits enhanced robustness against outliers and non-normal errors. We suggest a computationally efficient mode expectation-maximization algorithm leveraging a Gaussian kernel to numerically estimate the coefficient. Under mild assumptions, we derive the asymptotic distributions of the resulting kernel mode-based estimator, assuming that \(k_n\) increases to infinity at a slower rate than n. Specifically, for \(c<0\) , we establish a convergence rate of \(\sqrt{nk_n}\) with a normal limit distribution, while for \(c>0\) , the convergence rate is \(k_n \rho ^n_n\) with a Cauchy limit distribution. Monte Carlo simulations are presented to illustrate the favorable finite sample performance of the proposed estimation procedure. Furthermore, we extend these results to the general autoregressive process with a coefficient satisfying \(n |1-\rho _n |\rightarrow \infty \) under weaker initial conditions. The convergence rates are demonstrated to be \([n(1-\rho ^2_n)^{-1}]^{1/2}\) and \(\rho ^n_n/(\rho ^2_n-1)\) for nearly stationary and mildly explosive cases, respectively.
