A unified derivation of the complementary waiting time distribution in sequential occupancy
Occupancy distributions are defined on a stochastic model of random allocations of balls to a specific number of distinguishable urns. The reduction of the joint distribution of the occupancy numbers, when a specific number of balls is allocated, to the joint conditional distribution of independent random variables given their sum, when the number of balls allocated is unspecified, is a powerful technique in the study of occupancy distributions. Consider a supply of balls randomly distributed into n distinguishable urns and assume that the number X of balls distributed into any specific urn is a random variable with probability function \(P(X =x) =q_x\), \(x= 0,1,\dots\) The probability function of the number \(L_r\) of occupied urns until \(r\) balls are placed into previously occupied urns is derived in terms of convolutions of \(q_x\), \(x= 0, 1,\dots\) and their finite differences. Further, using this distribution, the minimum variance unbiased estimator of the parameter n, based on a suitable sequential sampling scheme, is deduced. Finally, some illustrating applications are discussed.
scientific article; zbMATH DE number 5119592