Under suitable regularity conditions, the order of consistency, i.e. the rate of convergence of consistent estimator is \(\sqrt {n}\) for a size n of sample. On the other hand, for a truncated family of distributions as a typical case when the regularity conditions do not necessarily hold, the order of consistency is not always \(\sqrt {n}\) , but there appear \(n^{1/\alpha }\) \((0<\alpha <2)\) , \((n\log n)^{1/2}\) , etc. In this chapter, using the variational distance and the amounts of information, the maximum order (or the bound for the order) of consistency is obtained and shown to be attainable.