Multivariate geometric distributions generated by a cumulative damage process
Esary, James Daniel
Marshall, A. W.
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Two (narrow and wide) multivariate geometric analogues of the Marshall-Olkin multivariate exponetial distribution are derived from the following cumulative damage model. A set of devices is exposed to a common damage process. Damage occurs in discrete cycles. On each cycle the amount of damage is an independent observation on a nonnegative random variable. Damages accumulate additively. Each device has its own random breaking threshold. A device fails when the accumulated damage exceeds its threshold. Thresholds are independent of damages, and have a Marshall-Olkin multivariate exponential distribution. The joint distribution of the random numbers of cycles up to and including failure of the devices has the wide multivariate geometric distribution. It has the narrow multivariate geometric distribution if the damage variable is infinitely divisible. (Author)
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NPS Report NumberNPS55EY73041A
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Esary, James Daniel; Marshall, A. W. (Monterey, California. Naval Postgraduate School, 1973-07); NPS55EY73071AA fairly common failure model in a wide variety of contexts is a cumulative damage process, in which shocks occur randomly in time and associated with each shock there is a random amount of damage which adds to previously ...
Esary, James D.; Marshall, Albert W. (Monterey, California: Naval Postgraduate School, 1970-09); NPS55EY7OO91AThe multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more ...
Esary, James D.; Marshall, Albert W. (Institute of Mathematical Statistics, 1974-01);The multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more ...