Design-Adjusted Parameter Estimation

This is the authors' abstract. We don't add key points for this paper.

SUMMARY The problem of adjusting for the effect of sample design in parametric estimation is examined and a general model-based approach to parametric inference with sample survey data is proposed. This approach includes, as a special case, recent results on model-based regression estimation with sample survey data. How does one adjust for the effect of sample design when estimating superpopulation model parameters from survey data? Recent papers by Nathan and Holt (1980), Holt, Smith and Winter (1980) and Holt and Scott (1981) have explored this problem from a model-based viewpoint when estimating regression parameters from sample survey data. The traditional way of dealing with this problem uses a design-based (randomisation theory) argument, which precludes any direct reference to a model. Essentially, in this case, one estimates a finite population parameter, which itself is an estimator of the superpopulation parameter of interest. See Kish and Frankel (1974) and Binder (1983). This paper describes a general approach to model-based inference with survey data which includes, as a special case, the results on model-based regression estimation referred to above. It also embodies a philosophy which turns out to be close to that underlying the design-based ap- proach to survey inference. The perceived differences between the model-based and design- based approaches in this area may not therefore be as wide as is sometimes believed to be the case. The suggested model-based approach is as follows. Suppose a finite population of N elements is generated by a stochastic process which involves a (possibly vector-valued) parameter 0. If observations could be made on all N elements of that population, the optimal estimator of 0 based on all these observations would usually be that value which minimised an appropriately chosen population level loss function. That is, a loss function for 0 defined in terms of data for all N elements in the population. This optimal estimator of 0 is then the same as the finite population parameter which design-based theory aims to estimate. Now suppose a sample can be used to estimate the values of that population loss function. The value which minimises this estimated population loss function can then be taken to be the optimal sample estimate of 0. This sample estimate is clearly also an estimate of the above finite population parameter. So far, the argument is only driving the inference problem one step back - from direct estimation of the superpopulation parameter itself to estimation of the population loss function. But, if the underlying stochastic process involves a matrix Z of known supplementary population values and a parameter A which characterises the conditional distribution of the