## Abstract

Several results that characterize the distribution of a lifetime variable. T, with probability mass function (pmf) p,, where t = 0, 1, 2, ..., by its survivor function, S-t = Sigma(jgreater than or equal tot)p(j) its hazard function. h(t) = p(t)/S-t, its cumulative hazard function, Lambda(t) = - ln S-t, its accumulated hazard function, H-t = Sigma(j=0)(t) h(t), and its mean residual life function, L-T = E[(T - t)\T greater than or equal to t] (an initially faulty item is deemed to have a zero lifetime), are presented. These include results that have previously appeared in the literature as well as some new results. Differences in the terminology used by engineers. actuaries. and biostatisticians are pointed out and clarified. Attention is focussed on the relationships between the IFR/DFR, IFRA/DFRA, NBU/NWU. NBUE/NWUE, and IMRL/DMLR classes to which a discrete lifetime distribution and its current age distribution belong.

Original language | English |
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Pages (from-to) | 3069-3093 |

Number of pages | 25 |

Journal | Communications in Statistics: Theory and Methods |

Volume | 33 |

DOIs | |

Publication status | Published - 2004 |

## Keywords

- reliability
- characterizations
- discrete distributions
- survivor function
- failure rate
- hazard function
- cumulative hazard function
- accumulated hazard function
- mean residual life function
- logconvexity
- logconcavity
- current age distribution
- TIME