Journal of Official Statistics, Vol.8, No.2, 1992. pp. 183200

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Weighting for Unequal Pi

Four distinct sources for unequal selection probabilities Pi of elements are distinguished concerning their origins, their effects, and their need for weights ki∝1/Pi. Three other types of weighting for estimation are also identified. Survey sampling theory is for unbiased estimation with weights ki but model based theory is against. The main disadvantage of weighting is the increase in variances from S2/n to S2(1+Ck2)/n for weighted estimates y¯w, where Ck2 is the relvariance of the ki. This is balanced against the increase of the mean square error of the unweighted estimate y¯u from S2/n to (S2/n+Rky2Ck2S2), where RkyCkS is the bias=y¯uy¯w of y¯u. This comparison of the mean square errors is explored for reasonable choices between y¯w and y¯u. Very recently (1990–91) some compromises are being suggested, especially “trimming” extreme weights, and “shrinkage” estimators. The problem becomes difficult for multipurpose surveys, which are much more common than a single purpose y¯w.

Selection probabilities; unequal selections; selection biases; self-weighting.

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