# An alternative approach to linearization variance estimators. Demnati and Rao (2004) derive a unified theory for linearization variance estimation using weights. Let Î¸ be the population quantity of interest, and define the estimator ËÎ¸ to be a function of the

An alternative approach to linearization variance estimators. Demnati and Rao (2004) derive a unified theory for linearization variance estimation using weights. Let Î¸ be the population quantity of interest, and define the estimator Ë†Î¸ to be a function of the vector of sampling weights and the population values:
Î¸= g (w, y1, y2. . . yk ),
Where w= (w1. . . wN)T with wi the sampling weight of unit i (wi =0 if i is not in the sample), and yj is the vector of population values for the jth response variable.
Then a linearization variance estimator can be found by taking the partial derivatives of the function with respect to the weights. Let

Evaluated at the sampling weights wi. Then we can estimate V (Ë†Î¸) by

For example, considering the ratio estimator of the population total,

The partial derivative of Ë†Î¸ = g (w, x, y) with respect to wi is

For an SRS, finding the estimated variance of á¹«z gives (4.11).
Consider the post stratified estimator in Exercise 17.
- Write the estimator as á¹«post =g (w, y, x1. . . xL), where xli =1 if observation i is in post stratum l and 0 otherwise.
- Find an estimator of V (á¹«post) using the Demnati€“Rao (2004) approach.