**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 E_{M}(y_{k}) = α + β × z_{k} with a variance
V_{M}(y_{k}) = σ^{2}, where y_{k} are independent random variables with the population
values Y_{k} as their assumed realizations, α, β and σ^{2} are unknown parameters,
Z_{k} are known population values of z, and E_{M} and V_{M} refer respectively to the
expectation and variance under the model. A regression estimator is given by

t_hat_{reg} = t_hat_{HT} + b(T_{Z} - t_hat_{ZHT}) where t_hat_{HT} is
the Horwitz-Thompson estimator of the total T, b is the estimated slope coefficient, and T_{Z} and
t_hat_{ZHT} are the known population total of z and the HT estimator of T_{Z}, respectively.
See pages 97-101 in Lehtonen, R & Pahkinen, E. (2004)