Adaptive bandwidth selection within the Robbins–Monro MS-NAR estimator

Develop and integrate a local adaptive bandwidth selection h_n(y) into the restoration–estimation Robbins–Monro algorithm for nonparametric estimation of Markov Switching Non-linear Autoregressive processes, in order to address increased estimator variance near the support boundaries.

Background

The authors observe that the variance of the kernel regression estimators increases at the edges of the support due to sparse data and sensitivity to the fixed bandwidth choice. They propose mitigating this by combining their restoration–estimation Robbins–Monro algorithm with local adaptive bandwidth selection h_n(y).

While the paper establishes asymptotic normality for fixed bandwidths, the discussion notes that extending these results to adaptive bandwidths poses significant technical challenges and is beyond the paper’s scope, motivating this explicit open problem.