On a restricted occupancy model and its applications

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Abstract
With the multivariate hypergeometric distribution as a background certain occupancy distributions useful in practical applications are derived. More specifically it is assumed that a sample ofnindividuals is drawn from a population consisting ofmtypes withrindividuals in each type, (i) without replacement and (ii) by returning the selected individual in the population and with it another individual of the same type. The distributions of the numberZof distinct types observed in the sample are obtained in both cases in terms of the numbers. Assuming, in addition to themequiprobable types of individuals, the existence of a control type, say, withsindividuals, the joint distribution of the numberUof distinct types observed in the sample and the numberVof individuals of the control type present in the sample is obtained in terms of the numbersC(n, k, r) and the marginal distribution ofUin terms of the Gould‐Hopper numbers. Using these distributions minimum variance unbiased estimators of the numbermof types are derived. Moreover small sample tests based on the zero frequency are constructed.