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Post-sampling efficient QR-prediction in large-sample surveys. (English) Zbl 0624.62016

The author generalized the concept of QR-predictors [R. L. Wright, J. Am. Stat. Assoc. 78, 879-884 (1983; Zbl 0535.62013)] originally proposed for diagonal covariance matrices of regression superpopulation models \(Y=X\beta +\epsilon\). The generalization concerns non-diagonal regression models provided fixed sampling design. The key idea consists in dealing with additional weights, \(r_ i\), \(q_{ij}\), that enable to modify both the estimation of \(\beta\), namely \({\hat \beta}=(X_ s'Q_ sX_ s)^{-1}X_ s'Q_ sY_ s\) (the index s indicates the sub-matrix or sub-vector corresponding to the actual sample), and of population total \[ T_{QR} = \bar x'{\hat \beta}+\sum_{s}r_ i(Y_ i-x_ i'{\hat \beta})/N. \] Examples show that a lot of usual estimators are special cases of this \(T_{QR}.\)
In the next section, asymptotic design-unbiased (ADU) prediction is studied. First, \(T_{QR}\) is proven to be ADU under relatively general conditions. Second, sufficient conditions are stated for a QR-predictor to be equivalent to a predictor of the general regression type. These conditions are always fulfilled for diagonal models.
Another section contains considerations on choice of the “best” QR- predictor. It is divided into two parts: design-based approach and model- based approach. In both cases only partial solutions are presented.
Reviewer: J.Herzmann

MSC:

62D05 Sampling theory, sample surveys

Citations:

Zbl 0535.62013
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