在本文中，我們證明了：若對應於壽命分佈函數F之更新函數為凸（ 凹

）族，因此解決了Shaked和Zhu(1992)所提出的兩個問題。 蝻う漫宒銵A

我們進一步得到了於某些特定之壽命分佈函數族中，

We prove that if the renewal function M(t) corresponding to a life distribution F is convex(resp. concave) then F is NBU(resp. NWU), and hence answer two questions posed by Shaked and Zhu(1992). Moreover, based on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Key words and phrases:exponential distribution;renewal counting process; DFR;NBU;NWU;NBUE;NWUE.

Barlow, R. E. and Proschan, F. (1964) Comparison of replacement policies, and renewal theory
implications. Ann. Math. Statist. 35,577-589.
Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Life Testing: Probability Models, 2nd
edn. To Begin With, Silver Springs, MD.
Brown, M. (1980) Bounds, inequalities, and monotonicity properties for some specialized renewal
processes. Ann. Probab. 8, 227-240.
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic
Press, New York.
Lau, K. S. and Rao, B. L. C. P. (1990) Characterization of the exponential distribution by the
relevation transform. J. Appl. Probab. 27, 726-729.
Marshall, A. W. and Proschan, F. (1972) Classes of distributions applicable in replacement with
renewal theory implication. Proc. 6th Berkley Symp. Math. Statist. Probab. 1, 395-415.
Shaked, M. and Zhu, H. (1992) Some results on block replacement policies and renewal theory. J.
Appl. Probab. 29, 932-946.