Inference in the presence of unknown rates

The convergence rate of an estimator can vary when applied to datasets from different populations. As the population is unknown in practice, so is the corresponding convergence rate. In this article, we introduce a method to conduct inference on estimators whose convergence rates are unknown. Specifically, we extend the subsampling approach of Bertail, Politis, and Romano (1999) to situations where the convergence rate may include logarithmic components. This extension proves to be particularly relevant in certain statistical inference problems. To illustrate the practical relevance and implementation of our results, we discuss two main examples: (i) non parametric regression with measurement error; and (ii) intercept estimation in binary choice models. In each case, our approach provides robust inference in settings where convergence rates are unknown; simulation results validate our findings.