You can also download a PDF copy of this lecture.

Two Estimators of a Domain Total Revisited

We saw that for a simple random sampling design there are two estimators of \(\tau_d\): \[ \hat\tau_d = N_d\bar{y}_d \ \ \ \text{and} \ \ \ \hat\tau_d = \frac{N}{n}n_d\bar{y}_d. \] The first has smaller variance, although it requires knowing \(N_d\). How can we explain the difference in variance using what we know about ratio estimators?

Consider that \[ \bar{y}_d = \frac{\sum_{i \in \mathcal{S}} y_i'}{\sum_{i \in \mathcal{S}} x_i}, \] where \[ y_i' = \begin{cases} y_i, & \text{if the $i$-th element is from the domain,} \\ 0, & \text{otherwise}, \end{cases} \] and \[ x_i = \begin{cases} 1, & \text{if the $i$-th element is from the domain,} \\ 0, & \text{otherwise}. \end{cases} \] Also note that \(N_d = \tau_x = \sum_{i = 1}^N x_i\) and \(n_d = \sum_{i \in \mathcal{S}} x_i\). So we can write these estimators as \[ N_d\bar{y}_d = \tau_x\frac{\sum_{i \in \mathcal{S}} y_i'}{\sum_{i \in \mathcal{S}} x_i} = \tau_x\frac{\frac{1}{n}\sum_{i \in \mathcal{S}} y_i'}{\frac{1}{n}\sum_{i \in \mathcal{S}} x_i} = \tau_x\frac{\bar{y}'}{\bar{x}} \] and \[ \frac{N}{n}n_d\bar{y}_d = \frac{N}{n}n_d\frac{\sum_{i \in \mathcal{S}} y_i'}{\sum_{i \in \mathcal{S}} x_i} = \frac{N}{n}n_d\frac{\sum_{i \in \mathcal{S}} y_i'}{n_d} = \frac{N}{n}\sum_{i \in \mathcal{S}} y_i' = N\bar{y}'. \] And note that \(y_i'\) is “approximately proportional” to \(x_i\) since \(y_i'\) = 0 if \(x_i\) = 0. So now why does the estimator \(N_d\bar{y}_d\) tend to have a smaller variance than the estimator \((N/n)n_d\bar{y}_d\)?

Ratio Estimators as Adjusted Estimators

Consider two estimators of \(\mu_y\): \[ \hat\mu_y = \bar{y} \ \ \ \text{and} \ \ \ \hat\mu_y = \frac{\bar{y}}{\bar{x}}\mu_x. \] Writing the ratio estimator as \[ \hat\mu_y = \frac{\mu_x}{\bar{x}}\bar{y} \] shows more clearly that the ratio estimator “adjusts” \(\bar{y}\) by a factor of \(\mu_x/\bar{x}\). \[\begin{align*} \bar{x} < \mu_x & \Rightarrow \frac{\mu_x}{\bar{x}}>1 \Rightarrow \frac{\mu_x}{\bar{x}}\bar{y} > \bar{y} \ \ (\text{i.e., adjust estimate up}) \\ \bar{x} = \mu_x & \Rightarrow \frac{\mu_x}{\bar{x}}=1 \Rightarrow \frac{\mu_x}{\bar{x}}\bar{y} = \bar{y} \ \ (\text{i.e., no adjustment}) \\ \bar{x} > \mu_x & \Rightarrow \frac{\mu_x}{\bar{x}}<1 \Rightarrow \frac{\mu_x}{\bar{x}}\bar{y} < \bar{y} \ \ (\text{i.e., adjust estimate down}) \end{align*}\] The factor of \(\mu_x/\bar{x}\) tells us if \(\mu_x\) is underestimated or overestimated by \(\bar{x}\). This gives us some idea that might have underestimated or overestimated \(\mu_y\) as well, so we might then adjust our estimate.

Example: Here \(\mu_x\) is underestimated by \(\bar{x}\). Example: Here \(\mu_x\) is overestimated by \(\bar{x}\).

Performance of Ratio Estimators

How does the relationship between the target and auxiliary variable affect the ratio estimator?

Example: In each of the following populations \(N\) = 1000 and \(\mu_y\) = 300.

Consider the sampling distributions of the ratio estimator \(\hat\mu_y = \mu_x\bar{y}/\bar{x}\) with \(n\) = 25.

How does the relationship between the target and auxiliary variable affect the ratio estimator, and how does this compare to using the “non-ratio” estimator? Is a ratio estimator always better than a “non-ratio” estimator? Can a ratio estimator be worse?

Example: Consider a population of \(N\) = 3000 elements (prisoners) where the target variable is finger length, and three estimators of \(\mu_y\):

  1. \(\hat\mu_y = \bar{y}\) (i.e., the sample mean)
  2. \(\hat\mu_y = \mu_h\bar{y}/\bar{h}\) (i.e., a ratio estimator using height as the auxiliary variable)
  3. \(\hat\mu_y = \mu_a\bar{y}/\bar{a}\) (i.e., a ratio estimator using age as the auxiliary variable)
The plots below show the distribution of finger length with height and with age in the population. The plots below show the sampling distributions of the three estimators based on a simple random sampling design with \(n\) = 25.
estimator variance B
Ratio Using Height 0.00174 0.08
Mean 0.00292 0.11
Ratio Using Age 0.07795 0.56

Sources of Auxiliary Variables for Ratio Estimators

  1. What is necessary for a variable to be used as an auxiliary variable?
  2. What is desirable for a variable to be used as an auxiliary variable?

What are some sources of auxiliary variables?

  1. Rough approximations to the target variable.
  2. Some measure of sampling unit size.
  3. Prior observations of the target variable from a census.