Demnati–Rao (2004) linearization variance estimator in two-phase sampling. The linearization variance estimator presented in Exercise 23 of Chapter 9 can be extended to two-phase sampling. Let θ be the population quantity of interest, and define the estimator ˆθ to be a function of the vectors of sampling weights for the phase I and phase II samples and the population values:

Where w(1) = (w(1)1 , . . . , w(1) N )T with w(1) i the phase I sampling weight of unit i (w(1)I = 0 if i is not in the phase I sample), w = (w1, . . . , wN)T with wi the final sampling weight of unit i in the phase II sample (wi = w(1)i w(2)i if i ∈ S(2) and wi = 0 if i ∈ S(2)), xj is the vector of population values for the jth auxiliary variable (measured in phase I), and yj is the vector of population values for the jth response variable (measured in phase II). Now let

And

Then,

- Consider the two-phase ratio estimator in (12.9).We can write

Where wi = w (1)i w(2)i . Show that the Demnati-Rao linearization variance estimator is

- Suppose that the phase I sample is an SRS of size n (1) and the phase II sample is an SRS of size n (2). What is ˆVDR (ṫ (2) yr) for this case?