An Asymptotic Distribution for an Occupancy Problem with Statistical Applications
M. Halperin, G. L. Burrows · Technometrics · 1961
Given a population of N items D of which are defective. Let the population be divided at random into k lots of equal size s (i.e. N = ks). This paper derives the asymptotic distribution (k → ∞) of the number of lots containing z or more defectives (x = 0, 1, 2, .…, s). Now suppose it is reasonable to define a lot as defective if it contains x or more defectives, and one is interested in the proportion of lots in a grand lot which are defective. Then on the basis of a random sample of items one can obtain confidence limits on various functions of the distribution of the number of defective lots. This allows high power against an excessive number of defective lots in an average sense (as in variables sampling) even for rather small samples. If, in fact, allocation of items to lots is not random but occurs with local positive contagion it appears that the random sample will tend to overstate the proportion of defective lots.