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July, 1973 Some Limit Theorems with Applications in Sampling Theory
Lars Holst
Ann. Statist. 1(4): 644-658 (July, 1973). DOI: 10.1214/aos/1176342460


From a finite population units are drawn with varying probabilities with replacement. There is a certain cost for observing a unit. In this paper samples are obtained partly by drawing a fixed number of times, and partly by drawing and observing units until the cost reaches a specified level. Let $X_k$ be the number of times the $k$th unit has been drawn in either case. Consider for a given function $g(\bullet)$ the random variable $Z = \sum_k g(X_k, k)$. Under general conditions it is proved that $Z$ is asymptotically normally distributed (actually a multidimensional generalization is considered). By appropriate choices of $g(\bullet)$ asymptotic distributions are obtained in successive sampling with varying probabilities without replacement and for the mean of the distinct units in a simple random sample with replacement. It is also investigated how heterogeneous catchability and effects of marking affect the "Petersen" estimator in capture-recapture theory.


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Lars Holst. "Some Limit Theorems with Applications in Sampling Theory." Ann. Statist. 1 (4) 644 - 658, July, 1973.


Published: July, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0259.62019
MathSciNet: MR365836
Digital Object Identifier: 10.1214/aos/1176342460

Keywords: 6030 , 6290 , capture-recapture estimation , limit theorems , Limit theorems in sampling theory , mean of distinct units , occupancy problems , sampling theory , sampling without replacement , successive sampling , unequal probability sampling

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 4 • July, 1973
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