Statistical inference based on judgment post-stratified samples in finite population Section 5. ExampleStatistical inference based on judgment post-stratified samples in finite population Section 5. Example

In this section we apply the proposed estimators to estimate corn production in Ohio based on 2012 United States Department of Agriculture (USDA) census. The population consists of $N =87$ counties in Ohio (One of the county is excluded from the population since census data did not have any entry for it). Variable of interest is the total corn production $(X)$ in bushels. We use 2007 USDA census corn production $(Y)$ as an auxiliary variable. Mean and standard deviation of corn production in 2012 are $μ_{X} = 5,021,061$ and $σ_{X} = 3,983,560$ bushels, respectively. The correlation coefficient between $X$ and $Y$ is 0.963. Using this population, we performed another simulation study to estimate the corn production and constructed confidence intervals for the population mean. Samples are generated for sample and set size combinations $(n, H) = (10,2), (15,3), (20,4) .$ Simulation and bootstrap replications sizes are taken to be 3,000 and 200, respectively. Rao-Blackwellized estimators are computed based on 50 replications.

Relative efficiencies of the estimators with respect to ${\tilde{μ}}_{2}$ and coverage probabilities of the confidence intervals are given in Table 5.1. Table 5.1 indicates that Rao-Blackwellized design-2 estimators outperforms all the other estimators we considered. Coverage probabilities appear to be slightly smaller than the nominal level 0.95.

Table 5.1
Relative efficiencies of estimators and coverage probabilities of a 95% confidence interval of population mean. The population is 87 Ohio counties. Variable of interest is corn production (X) in 2012. Auxiliary variable is corn production (Y) in 2007, $μ_{X} = 5,021,061,$ $σ_{X} = 3,983,560,$ $cor (X, Y) =0.963$ and $N =87$
Table summary
This table displays the results of Relative efficiencies of estimators and coverage probabilities of a 95% confidence interval of population mean. The population is 87 Ohio counties. Variable of interest is corn production (X) in 2012. Auxiliary variable is corn production (Y) in 2007, $μ_{X} = 5,021,061,$ $σ_{X} = 3,983,560,$ $cor (X, Y) =0.963$ and $N =87$ . The information is grouped by XXXXX (appearing as row headers), XXXXX, Relative Efficiencies, XXXXX and Coverage probabilities (appearing as column headers).
$n$	$H$	Relative Efficiencies, $R ({\bar{X}}_{0}) = Var ({\bar{X}}_{0}) / Var ({\tilde{μ}}_{2})$							Coverage probabilities
$n$	$H$	$R ({\bar{X}}_{0})$	$R ({\bar{X}}_{2})$	$R ({\hat{μ}}_{0})$	$R ({\hat{μ}}_{2})$	$R (μ_{0}^{*})$	$R (μ_{2}^{*})$	$R ({\tilde{μ}}_{0})$	$C^{a} ({\tilde{μ}}_{0})$	$C^{a} ({\tilde{μ}}_{2})$	$C^{b} ({\hat{μ}}_{0})$	$C^{b} ({\hat{μ}}_{2})$
10	2	2.301	1.981	1.829	1.448	1.468	1.280	1.181	0.883	0.896	0.924	0.925
15	3	3.745	3.188	2.353	1.612	1.994	1.454	1.200	0.907	0.919	0.940	0.907
20	4	5.707	4.402	2.901	1.624	2.476	1.143	1.341	0.920	0.920	0.946	0.873
$a :$ Coverage probabilities are computed from bootstrap percentile confidence interval. $b :$ Coverage probabilities are computed from ${\hat{μ}}_{r} \pm t_{n - 1, 0.975} {\hat{σ}}_{{\hat{μ}}_{r}},$ $r =0, 2.$

Table 5.2 presents the estimates of the standard deviation of the estimators of population mean from simulations and from analytic expression in equation (2.5), (2.6), (2.8), (3.2). It is again clear that estimates of the standard errors are reasonably close to the estimates from simulations. The standard deviation estimates of the estimators of the population total are obtained by multiplying the entries in Table 5.2 with the population size $N =87.$

Table 5.2
Estimates of the standard deviation of the estimators from 2012 USDA census. The population is 87 Ohio counties. Variable of interest is corn production (X) in 2012. Auxiliary variable is corn production (Y) in 2007, $μ_{X} = 5,021,061,$ $σ_{X} =3,983,560,$ $cor (X, Y) =0.963$ and $N =87$
Table summary
This table displays the results of Estimates of the standard deviation of the estimators from 2012 USDA census. The population is 87 Ohio counties. Variable of interest is corn production (X) in 2012. Auxiliary variable is corn production (Y) in 2007, $μ_{X} = 5,021,061,$ $σ_{X} =3,983,560,$ $cor (X, Y) =0.963$ and $N =87$ . The information is grouped by XXXXX (appearing as row headers), XXXXX, Estimates from equations (2.5), (2.6), (2.8), (3.2) and Estimates from simulation (appearing as column headers).
$n$	$H$	Estimates from equations (2.5), (2.6), (2.8), (3.2)					Estimates from simulation
$n$	$H$	${\hat{σ}}_{{\hat{μ}}_{0}}$	${\hat{σ}}_{{\hat{μ}}_{2}}$	${\tilde{σ}}_{{\hat{μ}}_{2}}$	${\hat{σ}}_{{\tilde{μ}}_{0}}$	${\hat{σ}}_{{\tilde{μ}}_{2}}$	$\sqrt{V^{a} ({\hat{μ}}_{0})}$	$\sqrt{V^{a} ({\hat{μ}}_{2})}$	$\sqrt{V^{a} ({\tilde{μ}}_{0})}$	$\sqrt{V^{a} ({\tilde{μ}}_{2})}$
10	2	1,108,818.7	1,027,289.0	1,027,717.4	883,847.8	833,711.4	1,156,300.5	1,028,629.9	929,090.9	854,940.3
15	3	815,371.3	687,605.0	689,118.9	602,682.4	545,000.2	810,156.1	670,521.9	578,608.4	528,146.5
20	4	652,734.4	472,231.5	477,888.6	454,368.3	392,990.3	638,755.1	478,007.6	434,365.0	375,040.7
$a :$ These variance estimates are obtained from simulation.

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Statistical inference based on judgment post-stratified samples in finite population Section 5. ExampleStatistical inference based on judgment post-stratified samples in finite population Section 5. Example