In this paper we present characterizations of the Pareto, Lomax, exponential and beta models by some properties of their $r^{th}$ partial moment defined as ${alpha}_r(t)=E[(X-t)^+]^r$, where $(X-t)^+ = max(X-t,0)$. Given the partial moments at a few truncation points, these results enable us to calculate the moments at many other points.