We establish a mathematical framework that formally validates the two-phase “superpopulation viewpoint” proposed by Hartley and Sielken (1975), by defining a product probability space which includes both the design space and the model space. We develop a general methodology that combines finite population sampling theory and classical theory of infinite population sampling to account for the underlying processes that produce the data. Key results in this article are: the sample estimator and the model statistic are asymptotically independent; if a sequence converges in design law, it also converges in the law of the product space; and the distribution theory of the sample estimating equation estimator around a super-population parameter. We also study the interplay between dependence and independence of random variables when viewed in the design space, the product space and the model space and apply it to show formally that under a “simple random sample without replacement” design, we can “ignore” the design and work on the realm of the model space, but that under “simple random sample with replacement” we cannot ignore the design.
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Paper provided by Département des sciences administratives, UQO in its series RePAd Working Paper Series with number
lrsp-TRS382.
Length: 29 pages Date of creation: 01 Jan 2001 Date of revision: Handle: RePEc:pqs:wpaper:0142005
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Find related papers by JEL classification: C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - General C40 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - General