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Biometrika 1968 55(1):229-241; doi:10.1093/biomet/55.1.229
© 1968 by Biometrika Trust
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Sequential occupancy with classification

D. E. BARTON and F. N. DAVID

University of London, Institute of Computer Science and Queen Mary College
University of California Riverside and Berkeley

A series of independent K-way Bernoulli trials is continued until a compound event E, contingent on the occurrences or non-occurrences of the K types of outcome, has occurred at the nth trial. A general form for the exponentially weighted probability generating function of n is obtained and corresponding moment formulae derived, in particular for the mean, which is important in applications. The special object of the study is when the K outcomes are hierarchically arranged or cross-classified, and, in the former case, the compound event is the occurrence of all the events in a ‘block’, for at least one of m blocks. In particular the mean of n for this case is shown to be closely K{log m—log log (K/m)}. When the trials are not independent but have a hypergeometric form, corresponding to sampling without replacement, then a parallel form of binomially weighted p.g.f. is obtained. A cross-classified model is treated where the stopping rule is when the first set of outcomes in a complete row or column have all appeared.


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