The convergence rate of an estimator can vary when applied to datasets from differ- ent populations. As the population is unknown in practice, so is the corresponding convergence rate. In this paper, we introduce a method to conduct inference on estimators whose conver- gence rates are unknown. Specifically, we extend the subsampling approach of Bertail, Politis, and Romano (1999) to situations where the convergence rate may include logarithmic compo- nents. This extension proves to be particularly relevant in economic studies. 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.