A method to correct for frame membership error in dual frame estimators
Section 2. Dual frame estimation

• Previous
• Next

Since the introduction of dual frame sample designs by Hartley (1962), many estimators have been proposed (Fuller and Burmeister, 1972; Kalton and Anderson, 1986; Bankier, 1986; Skinner and Rao, 1996). In this section, we focus on Hartley’s estimator, since it was used in our MRIP pilot study application. Using Hartley’s original notation, we denote by $N\mathrm{,}{N}_A\mathrm{,}{N}_B\mathrm{,}{N}_a\mathrm{,}{N}_b$ and $N_{ab}$ the number of elements in $U\mathrm{,}A\mathrm{,}B\mathrm{,}a\mathrm{,}b$ and $ab,$ respectively. Then the following relationships hold: $N\mathrm{<mo>=</mo>}{N}_a+N_{ab}+N_b,$ $N_A\mathrm{<mo>=</mo>}{N}_a+N_{ab},$ and $N_B\mathrm{<mo>=</mo>}{N}_{ab}+N_b.$ Denote the samples from frames $A$ and $B$ as $s_A$ and $s_B,$ and unit $i’\text{s}$ inclusion probability in the two samples as ${\pi }_i^A$ and ${\pi }_i^B.$

The population total $Y$ can be written as the sum of totals of the three mutually exclusive domains.

$$Y\mathrm{<mo>=</mo>}{Y}_a+Y_{ab}+Y_b\mathrm{,}(2.1)$$

where $Y_a\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in a}}{y}_i,$ $Y_b\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in b}}{y}_i$ and $Y_{ab}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in ab}}{y}_i.$ Estimators of the total can be written as the sum of total estimators in the three different domains, which is

$$\widehat{Y}\mathrm{<mo>=</mo>}{\widehat{Y}}_a+{\widehat{Y}}_{ab}+{\widehat{Y}}_b\mathrm{.}(2.2)$$

Hartley’s estimator (1962) is

$${\widehat{Y}}_H\mathrm{<mo>=</mo>}{\widehat{Y}}_a+\theta {\widehat{Y}}_{ab}^A+\left(1-\theta \right){\widehat{Y}}_{ab}^B+{\widehat{Y}}_b\mathrm{,}(2.3)$$

where ${\widehat{Y}}_{ab}^A\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_{ab}^A}}{w}_i^A{y}_i$ denotes the estimator of $Y_{ab}$ using information from frame $A,$ and ${\widehat{Y}}_{ab}^B\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_{ab}^B}}{w}_i^B{y}_i$ is the corresponding estimator from frame $B.$ $s_{ab}^A$ $\left(s_{ab}^B\right)$ is the subset of $s_A$ $\left(s_B\right)$ consisting of items that fall in domain $ab$ and $w_i^A\mathrm{<mo>=</mo>}1/{\pi }_i^A$ and $w_i^B\mathrm{<mo>=</mo>}1/{\pi }_i^B$ are the sampling weights based on the inclusion probabilities in frames $A$ and $B.$ In practice, those weights are typically adjusted for non-response and possibly undercoverage as well. $Y_a$ and $Y_b$ are estimated similarly as ${\widehat{Y}}_a\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_a}}{w}_i^A{y}_i$ and ${\widehat{Y}}_b\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_b}}{w}_i^B{y}_i.$ $\theta$ is a number between 0 and 1 that adjusts the weights of items from frames $A$ and $B.$ When $\theta$ denotes a constant, theory provides an optimal value for it that will minimize the estimator’s variance. However it will depend on unknown parameters, and thus must be estimated. Another approach that may be used is to select a value of $\theta$ that is proportional to the reciprocal of the sample sizes from the two frames. This will be a constant, and will be near optimal if the sample designs have similar design effects and a small overlap domain.

If $\theta$ is a constant, ${\widehat{Y}}_H$ is linear in the data, and its properties are easy to calculate. If $\theta \mathrm{<mo>=</mo>}0$ or 1, the estimator for the overlap domain depends on data from only one of the two frames. The optimal value of $\theta$ for minimizing the variance of ${\widehat{Y}}_H$ (Hartley, 1962) is

$${\theta }_H\mathrm{<mo>=</mo>}\frac{V\left({\widehat{Y}}_{ab}^B\right)+\text{Cov}\left({\widehat{Y}}_b^B\mathrm{,}{\widehat{Y}}_{ab}^B\right)-\text{Cov}\left({\widehat{Y}}_a^A\mathrm{,}{\widehat{Y}}_{ab}^A\right)}{V\left({\widehat{Y}}_{ab}^A\right)+V\left({\widehat{Y}}_{ab}^B\right)}\mathrm{.}(2.4)$$

The variances and covariances in (2.4) are unknown and must be estimated from the sample if the optimal form for $\theta$ is to be used in (2.3). In that case, the resulting weights are random and contribute to the variability of the estimator. Another disadvantage of using the optimal value of $\theta$ in ${\widehat{Y}}_H$ is that ${\theta }_H$ is different for different response variables, which results in inconsistency among estimators of related quantities. For example, the optimal estimators of the number of shore and boat fishing trips made by anglers do not necessarily sum to the optimal estimator of total number of fishing trips. For that reason, in practice, a constant value of $\theta$ is frequently used; for example, a value of $\theta \mathrm{<mo>=</mo>}1/2$ is sometimes recommended when sample sizes are similar in the two frames (Lohr, 2011).

• Previous
• Article main page
• Next