New better than used in expectation (${\int }_x^{\infty }{\overline{F}}_X(t)dt\le \mu {\overline{F}}_X(x)$)

A nonnegative lifetime or claim law is new better than used in expectation if its mean residual life at every age with positive survival is at most its original mean. The displayed equivalent inequality says that its equilibrium claim-size distribution is no larger in stochastic order than the original law. For an Erlang distribution of shape two and rate one, the two survival functions are $(1+x/2)e^{-x}$ and $(1+x)e^{-x}$, so the condition holds.

For NBUE claims in the classical risk model, the equilibrium claim size $Y$ is stochastically bounded by $X$. The positive exponential tail integral formula for moments gives $M_Y(R)\le M_X(R)$. Since $M_Y(R)=1+\theta$ and $M_X(R)=1+(1+\theta )\mu R$, rearrangement proves the displayed bound.