Exponential approximations in the classes of life distributions

Abstract

In this paper we discuss the approximation of life distributions by exponential ones. The main results are: (1) ∀F∈ NBUE, where its mean is 1, we have$|\bar F(t) - e^{ - t} | \leqslant 1 - e^{ - \sqrt {20} } ,\forall t \geqslant 0$, ∀≥0, where ρ = 1 - μ₂/2, μ₂ being the second moment ofF. The inequality is sharp. (2) In the case ofF∈IFR, the upper bound is$1 - e^{ - \tfrac{\rho }{{1 - \rho }}}$. (3) For the HNBUE class, the upper bound is min$(\sqrt[3]{{4\rho }}.\sqrt[3]{{4\rho }})$. Furthermore, the improved upper bound is$\sqrt[3]{{36\rho /(3 + 2\sqrt \rho )^2 }}$. In addition, we show$\mathop {\sup }\limits_{t > 0} |\bar G(t) - e^{ - t} | \leqslant \sqrt {\frac{\rho }{2}}$, where$\bar G(t) = \int_t^\infty {\bar F} (u)du$ (4) For the IMRL class, the upper bound is ρ/(1+ρ) ([1]). Here we give a simple proof.