A companion paper develops a framework in which probability measures are represented by distribution-kernel pairs (T,phi) with T a tempered distribution and phi a Schwartz kernel, so that weak moments of all orders exist unconditionally. The present paper turns this into a methodology for statistical inference: estimation via weak moment matching, weak characteristic functions, weak cumulants, and regularised density reconstruction via Tikhonov inversion. A key feature is that parametric inference proceeds directly from weak expectations without reconstructing the underlying density; reconstruction is an additional route, useful when density-level inference is the goal. The central result is that weak moment estimators are automatically locally robust in the sense of Hampel: their score is bounded and redescending, their influence function has a closed form, and their gross error sensitivity is finite in every identifiable parametric model -- all inherited from the kernel's decay, with no ad hoc truncation. The kernel plays the role of Huber's tuning constant, but as a structural component of the model rather than a post-hoc modification. The framework is worked out for the Cauchy location model (where no classical moment estimator exists), a Student t_3 location-scale model, a bivariate Cauchy location model, and a bivariate t_3 location-scale model. Monte Carlo comparisons show that weak moment estimators match or outperform classical robust benchmarks under contamination; in the bivariate t_3 case the MLE scale estimate breaks down while the weak moment estimator converges at the parametric rate. Although the paper focuses on parametric models, the reconstruction route is inherently non-parametric and opens a path to weak density estimation without parametric assumptions.