On dealing with the unknown population minimum in parametric inference

A myriad of physical, biological and other phenomena are better modeled with semi-infinite distribution families, in which case not knowing the population minimum becomes a hassle when performing parametric inference. Ad hoc methods to deal with this problem exist, but are suboptimal and sometimes unfeasible. Besides, having the statistician handcraft solutions in a case-by-case basis is counterproductive. In this paper, we propose a framework under which the issue can be analyzed, and perform an extensive search in the literature for methods that could be used to solve the aforementioned problem; we also propose a method of our own. Simulation experiments were then performed to compare some methods from the literature and our proposal. We found that the straightforward method, which is to infer the population minimum by maximum likelihood, has severe difficulty in giving a good estimate for the population minimum, but manages to achieve very good inferred models. The other methods, including our proposal, involve estimating the population minimum, and we found that our method is superior to the other methods of this kind, considering the distributions simulated, followed very closely by the endpoint estimator by Alves et al. (Stat Sin 24(4):1811–1835, 2014). Although these two give much more accurate estimates for the population minimum, the straightforward method also displays some advantages, so choosing between these three methods will depend on the problem domain.