VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two-phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s(1) is drawn according to a specific sampling design p(s(1)), and auxiliary data x are observed for the units i is an element of s(1), Given the first-phase sample s(1), a second-phase sample s(2) is selected from s(1) according to a specified sampling design {P (s(2) vertical bar s(1))}, and (y, x) is observed for the units i is an element of s(2). In some cases, the population totals of some components of x may also be known. Two-phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz-Thompson-type variance estimators are used for variance estimation. However, the Horvitz-Thompson (Horvitz & Thompson, J. Amer. Statist. Assoc. 1952) variance estimator in uni-phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen-Yates-Grundy variance estimator is relatively stable and non-negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen-Yates-Grundy (Sell, J. Ind. Soc. Agric. Statist. 1953: Yates & Grundy, J. Roy. Statist. Sic. Ser: B 1953) variance estimator to two-phase sampling, assuming fixed first-phase sample size and fixed second-phase sample size given the first-phase sample. We apply the new variance estimators to two-phase sampling designs with stratification at the second phase or both phases. We also develop Sen-Yates-Grundy-type variance estimators of the two-phase regression estimators that snake use of the first-phase auxiliary data and known population totals of some of the auxiliary variables.