A Lower Bound on the L -Class of Life Distributions and its Applications

Abstract

Two life distributions F and G can be ordered with respect to their Laplace transforms. We say that F is Laplace­larger than G if $\underset{o}{\overset{\infty }{\int }}\hspace{0.17em}\exp (\hspace{0.17em}-st)\hspace{0.17em}\overline{F}(t)\hspace{0.17em}dt\hspace{0.17em}⩾\hspace{0.17em}\underset{o}{\overset{\infty }{\int }}\hspace{0.17em}\exp (\hspace{0.17em}-st)\hspace{0.17em}\overline{G}(t)\hspace{0.17em}dt$ for all s ⩾ 0, where $\overline{F}=\hspace{0.17em}1\hspace{0.17em}-\hspace{0.17em}F$ . Let F be a distribution function with finite mean µ. Then F is said to belong to the L-class of life distributions if F is Laplace-larger than G, where G is exponential with the same mean µ. In this paper we obtain a lower bound on $\overline{F}\hspace{0.17em}\epsilon \hspace{0.17em}L$ in terms of µ. Some applications are indicated through examples.