Composite Estimation

How a Composite Estimator Works

In a typical rotation design, the sampled groups are phased in and out of the sample in a regular, denned pattern over time. To estimate the level of a characteristic in the time period designated by t, a simple compositing strategy is to take a convex combination of the Horvitz-Thompson estimate of level for period t, YHTlt, with a second estimate for period t, YHT2t The latter estimate might start with the composite estimate for period t − 1, YCEt − 1 brought forward by a measure of change from period t − 1 to period t:

This measure of change, Δt − 1,t, can be a difference (ratio) estimated using data only from the overlapping [Page 116]rotation groups, which is then added to (multiplied by) the composite estimate for period t—l. The composite estimate then becomes a recursively denned function of data collected in prior time periods:

where 0 < k < 1.

Composite estimators can often be expressed as a linear combination of simple estimates—one formed from each rotation group at each period. A few constraints are usually imposed. First, when estimating levels of a variable at time t, one usually requires that (a) the weighting coefficients of the group estimates at time t add to 1, and (b) for each period before t, the coefficients sum to 0. These restrictions ensure that no bias is introduced through the compositing. Second, to maintain the consistency of estimates, it is customary, at least for statistical agencies, to require that (a) the estimate of changes in a variable equal the difference (or ratio, for multiplicative composite estimators) of the appropriate estimates of levels for that variable, and (b) the estimates of components sum to the estimate of the corresponding total.

Composite estimation tries to take advantage of correlations over time. For example, suppose xt − 1,g and xt,g are estimates from the same rotation group, g, for periods t − 1 and t. If, due to sampling variability, xt − 1,g is below its expected value, then xtg tends to be as well. By assigning coefficients with opposite signs to the two estimates, one can temper the sampling variations while still balancing coefficients to ensure an unbiased estimate overall.

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