Spurious exponentiality observed when incorrectly fitting a distribution to nonstationary data

Failure data for a repairable system can be represented either by a set of chronologically ordered arrival times at which the system failed, or by a set of interarrival times defined as the times observed between successive failures (ignoring repair times in both cases). The two representations are mathematically equivalent if the chronological order of the interarrival times is maintained. Methods aimed at describing the distribution of the observed interarrival times are meaningful only if the interarrival times are identically distributed. In all other cases, such analyses are meaningless and often result in maximally misleading impressions about the system behavior, as demonstrated here by several examples. That is, when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process. Misunderstandings of this nature can be avoided by applying an appropriate test for trend before attempting to fit a distribution to the interarrival times. If evidence of trend is determined, then a nonstationary model such as the nonhomogeneous Poisson process should be fitted using the chronologically ordered data.