Ageing Properties of the Time to Extinction of a Multi­Type Branching Process

Abstract

Bhattacharjee (1987) and Shaked and Shantikumar (1987) have shown that the distribution of the time to extinction T of a sub­critical or a critical Galton­ Watson branching process has a DFR property. We give an example to show that such a property does not hold for a multi­type Galton­ Watson process. ln this note, we further study properties of T for a multitype process. We show that its ultimate failure rate is a constant. In the sub­critical case, it is also shown that its distribution is smaller than the geometric distribution (whose failure rate is the above limiting failure rate), in the sense of likelihood ratio ordering. We then show that the distribution of T in the sub­critical case belongs to the NWOE (New Worse than Old in Expectation) class of distributions.