
Weighting for Unequal P_{i} Leslie Kish Abstract: Four distinct sources for unequal selection probabilities P_{i} of elements are distinguished concerning their origins, their effects, and their need for weights k_{i}∝1/P_{i}. Three other types of weighting for estimation are also identified. Survey sampling theory is for unbiased estimation with weights k_{i} but model based theory is against. The main disadvantage of weighting is the increase in variances from S^{2}/n to S^{2}(1+C_{k}^{2})/n for weighted estimates y¯_{w}, where C_{k}^{2} is the relvariance of the k_{i}. This is balanced against the increase of the mean square error of the unweighted estimate y¯_{u} from S^{2}/n to (S^{2}/n+R_{ky}^{2}C_{k}^{2}S^{2}), where R_{ky}C_{k}S is the bias=y¯_{u}−y¯_{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}. Keywords: Selection probabilities; unequal selections; selection biases; selfweighting.
