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.
Publié le : 1973-07-14
Classification:
6030,
6290,
Limit theorems in sampling theory,
limit theorems,
sampling theory,
sampling without replacement,
successive sampling,
mean of distinct units,
unequal probability sampling,
occupancy problems,
capture-recapture estimation
@article{1176342460,
author = {Holst, Lars},
title = {Some Limit Theorems with Applications in Sampling Theory},
journal = {Ann. Statist.},
volume = {1},
number = {2},
year = {1973},
pages = { 644-658},
language = {en},
url = {http://dml.mathdoc.fr/item/1176342460}
}
Holst, Lars. Some Limit Theorems with Applications in Sampling Theory. Ann. Statist., Tome 1 (1973) no. 2, pp. 644-658. http://gdmltest.u-ga.fr/item/1176342460/