CHARACTERIZING A NEGATIVE BINOMIAL PROCESS FOR A GAMMA DISTRIBUTED FAILURE RATE

Abstract

Used as a mixing distribution for a random Poisson parameter, the gamma distribution leads to a negative binomial process. This appears to be a useful model for failure data, particularly for data from a number of repairable systems all of which follow a Poisson process but with different intensities. The hyper-parameters of the gamma distribution have different meanings according to the sources of randomness in the Poisson failure parameter. Two such sources are failure time and failure rate. Random failure time and random failure rate are interpreted in the resulting negative binomial average failure in terms of the number of failures and the intensity of a failure, respectively.