On combining independent probability samples
Section 4. Simulation examples

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Two Monte-Carlo simulation examples are presented here. In the first example we combine two Poisson samples using inclusion probabilities approximately proportional to the target variable. In the second example we combine an unstratified simple random sample with a stratified simple random sample.

4.1  Combining two Poisson samples

We generate a population of size $N\mathrm{<mo>=</mo>\; 200},$ with an auxiliary variable $X_i\sim N\left(\mu =\mathrm{20,}{\sigma }^2=16\right).$ The target variable is generated as $\left(Y_i|X_i\mathrm{<mo>=</mo>}{x}_i\right)\mathrm{<mo>=</mo>}{x}_i+{ϵ}_i,$ where ${ϵ}_i\sim N\left(\mathrm{0,}{\left(x_i/20\right)}^2\right).$ Two sets of inclusion probabilities are generated ${\pi }_i^{\left(1\right)}\propto {\pi }_i^{\left(2\right)}\propto x_i,$ where ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{\left(1\right)}\mathrm{<mo>=</mo>}{n}_1}$ and ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{\left(2\right)}\mathrm{<mo>=</mo>}{n}_2}.$ We let the expected sample sizes be $n_1\mathrm{<mo>=</mo>\; 15}$ and $n_2\mathrm{<mo>=</mo>\; 25.}$ For simplicity we let both designs be Poisson designs (where units are selected independently). This allows us to calculate exactly the variances for both separate estimators (and thus the optimal linear combination) and for the combined samples with single and multiple count. For the strategies with linear combination using estimated variances, we performed a Monte-Carlo simulation with 1,000,000 repeated sample selections. True variances for the two separate HT estimators, the SC/MC estimators for the combined samples and the optimal linear combination of the separate estimators are presented (Table 4.1). Simulation results for the different linear combinations with estimates variances are also presented (Table 4.1).

Using combined (pooled) variance estimators reduced both the bias and the variance for a linear combination in comparison to using separate variance estimators. For this example, the linear combination with pooled variance estimation came very close to the optimal linear combination in performance. The negative bias with separate variance estimators is mainly due to a positive correlation between the total estimator and its variance estimator under the Poisson design. For this setting, the best result was obtained by combining the samples using a single count.

4.2  Combining an unstratified SRS with a stratified SRS

Here we generated a population of size $N\mathrm{<mo>=</mo>}\text{1,000},$ with two strata of sizes $N_1\mathrm{<mo>=</mo>\; 600}$ and $N_2\mathrm{<mo>=</mo>\; 400.}$ The target variable $y_i,$ $i\mathrm{<mo>=</mo>\; 1,}\dots ,N,$ was generated as follows. In stratum 1 there were 500 $y_i’\text{s}$ equal to zero and the other 100 $y_i’\text{s}$ were drawn from $N\left({\mu }_1\mathrm{<mo>=</mo>\; 10,}{\sigma }^2\mathrm{<mo>=</mo>\; 4}\right).$ In stratum 2 there were 300 $y_i’\text{s}$ equal to zero and the other 100 $y_i’\text{s}$ were drawn from $N\left({\mu }_2\mathrm{<mo>=</mo>\; 15,}{\sigma }^2\mathrm{<mo>=</mo>\; 4}\right).$ The first sample is an unstratified simple random sample of size $n\mathrm{<mo>=</mo>\; 50}$ and the second sample is a stratified simple random sample with stratum sample sizes $n_1\mathrm{<mo>=</mo>\; 30}$ and $n_2\mathrm{<mo>=</mo>\; 20.}$ The variances for both separate HT estimators and for the combined samples with single and multiple count were calculated exactly. A Monte-Carlo simulation with 10,000 repetitions was performed to evaluate the performance of a linear combination estimator with estimated variances. The results are presented in Table 4.2. Bias is reduced by using a linear combination with pooled variance estimators compared with using separate variance estimators. Also for this setting, the best result was obtained by combining the samples using a single count.

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