Regression estimation of totals

Regression estimation of the population total T of a study variable y is based on the linear regression between y and a continuous auxiliary variable z. The linear regression can, for example, be given by EM(yk) = α + β zk with a variance VM(yk) = σ2, where yk are independent random variables with the population values Yk as their assumed realizations, α, β and σ2 are unknown parameters, Zk are known population values of z, and EM and VM refer respectively to the expectation and variance under the model. A regression estimator is given by
t_hatreg = t_hatHT + b(TZ - t_hatZHT) where t_hatHT is the Horwitz-Thompson estimator of the total T, b is the estimated slope coefficient, and TZ and t_hatZHT are the known population total of z and the HT estimator of TZ, respectively. See pages 97-101 in Lehtonen, R & Pahkinen, E. (2004)

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