Estimation when the Variable of Interest is Correlated

What is it about?

Generalized regression (GREG) estimation uses a model which assumes the values of the variable of interest, are not correlated. An extension of the GREG estimator to the case where the vector of interest has a positive definite covariance structure is presented in this paper. For example, secondary units that belong to the same primary unit are often correlated. This extension can be translated to the calibration estimators. The key to this extension lies in a generalization of the Horvitz-Thompson estimator. The Godambe-Joshi lower bound is another result which assumes a model with no correlation. This is also generalized to a vector of interest with a positive definite covariance structure, and it is shown that the generalized calibration estimator asymptotically attains this generalized lower bound. Properties of the new estimators are given, and they are compared to the Horvitz-Thompson estimator and the usual calibration estimator. The new estimators are applied to the problem of variance estimation.

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Why is it important?

If the sampling units are correlated, then the ubiquitous Horvitz-Thompson estimator can be improved upon. The GREG estimator and the calibration estimator are no longer optimal if the units are correlated. Estimators that take the correlation into account are shown to perform better.