Asymptotic Distribution for a Generalized Banach Match Box Problem
T. Cacoullos · Journal of the American Statistical Association · 1967
Balls are drawn one after another from k cells C 1, …, Ck according to the multinomial distribution. Suppose the ith cell initially contains Ni balls, and sampling stops as soon as any of the k cells, say Cα , empties first. Let Xi denote the number of balls taken from cell Ci (all i ≠ α) at stopping time. The joint asymptotic (as Ni → ∞) distribution of the Xi (i ≠ α) is derived under the most general configuration of the multinomial cell probabilities p 1, …, pk. Conditions on pi and Ni are given under which the asymptotic distribution is shown to be either normal or truncated (restricted) normal. An application of the asymptotic distribution theory for Ni = N 0 (i = 1, …, k) in setting up approximate tests and confidence intervals for the largest pi is also given. Under certain conditions on pi and Ni it is shown that the asymptotic probability that Ci empties first is equal to the probability content of a positive orthant under a multivariate normal distribution.