Probability-proportional-to-size ranked-set sampling from stratified populations
Section 5. Efficiency comparison of the new sampling design and estimator

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In this section, we investigate the efficiency of the stratified-PPS-ranked-set sample estimators. We consider a stratified population with three strata $\left(L\mathrm{<mo>=</mo>\; 3}\right).$ To see the effects of the stratum population sizes and variances on the allocation procedures, we generated stratified populations with different population sizes and variances. For clarity of notation, we define the proportions of population sizes and variances as follows:

$$p_{N_l}=\frac{N_l}{{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>\; 1}}^L{N}_l}}\mathrm{,}{p}_{{\sigma }_{N_l}}\mathrm{<mo>=</mo>}\frac{{\sigma }_{N_l}}{{\displaystyle {\sum }_{l=1}^L{\sigma }_{N_l}}}\mathrm{,}l\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}L\mathrm{.}$$

We note that proportional and Neyman allocations select stratum sample sizes proportional to $p_{N_l}$ and $p_{{\sigma }_{N_l}},$ respectively.

In this part of the simulation, the population values of the $Y$ - and $X$ -variables are generated with a model different from the models in equations (2.3) and (2.4). For stratum population $l,$ we generate $X_l^{*}\mathrm{<mo>=</mo>}\left(X_{\mathrm{1,}l}^{*}\mathrm{,}\dots \mathrm{,}{X}_{N_l\mathrm{,}l}^{*}\right)$ from

$$X_{i\mathrm{,}l}^{*}\mathrm{<mo>=</mo>}{F}^{-1}\left(i/\left(N+1\right)\mathrm{<mo>;</mo>\; 0.1}\right)\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_l\mathrm{,}$$

where $F\left(\mathrm{<mo>;</mo>}\lambda \right)$ is the cumulative distribution function of the exponential distribution with mean 10 (rate $\lambda \mathrm{<mo>=</mo>}$ 0.1). To simplify construction of the values of the $X$ -variable from the stratum population, we re-scaled $X_{i\mathrm{,}l}^{*}$ by

$$X_{i\mathrm{,}l}=\sqrt{\frac{X_{i\mathrm{,}l}^{*}}{\text{min}\left(X{}_l^{*}\right)}}\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_l\mathrm{.}$$

The values of the variable $Y$ in the stratum population $l$ are generated from the quantiles of a normal distribution using the variable $X_{i\mathrm{,}l},$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_l.$ We first compute

$${\epsilon }_{i\mathrm{,}l}\mathrm{<mo>=</mo>}{G}^{-1}\left(i/\left(N_l+1\right)\mathrm{<mo>;</mo>}{\theta }_l\mathrm{,}{\tau }_l\right)\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_l\mathrm{,}$$

where $G\left(\mathrm{<mo>;</mo>}a\mathrm{,}b\right)$ is the CDF of a normal distribution with mean $a$ and standard deviation $b.$ The values of the $Y$ -variable are then constructed from

$$Y_{i\mathrm{,}l}\mathrm{<mo>=</mo>}{X}_{i\mathrm{,}l}{\epsilon }_{i\mathrm{,}l}\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_l\mathrm{.}$$

In this construction, it is clear that the values of the $Y$ -variable are proportional to the values of the $X$ -variable. Hence, use of the stratified-PPS-ranked-set sample would be appropriate.

For the simulation study, the total population size $N,$ the sample size $n$ and the location parameter ${\theta }_l\left(l\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}3\right)$ are selected to be $N\mathrm{<mo>=</mo>}$ 700, $n\mathrm{<mo>=</mo>}$ 90, and ${\theta }_l\mathrm{<mo>=</mo>}$ 5, respectively. The values $N_l$ and ${\tau }_l$ are varied to establish the values of $p_{N_l}$ and $p_{{\sigma }_{N_l}}$ in Tables 5.1 and 5.2. For the first four rows, the population sizes and standard deviations are selected to be $N_1\mathrm{<mo>=</mo>}$ 100, $N_2\mathrm{<mo>=</mo>}$ 200, $N_3\mathrm{<mo>=</mo>}$ 400 and ${\sigma }_{N_1}\mathrm{<mo>=</mo>}$ 35, ${\sigma }_{N_2}\mathrm{<mo>=</mo>}$ 10, ${\sigma }_{N_3}\mathrm{<mo>=</mo>}$ 5, respectively. For the last four rows, the population sizes and standard deviations are selected as $N_1\mathrm{<mo>=</mo>}$ 400, $N_2\mathrm{<mo>=</mo>}$ 200, $N_3\mathrm{<mo>=</mo>}$ 100 and ${\sigma }_{N_1}\mathrm{<mo>=</mo>}$ 35, ${\sigma }_{N_2}\mathrm{<mo>=</mo>}$ 10, ${\sigma }_{N_3}\mathrm{<mo>=</mo>}$ 5, respectively. To make the comparison easier, the same set size $H$ $\left(H\mathrm{<mo>=</mo>}\mathrm{2,}\mathrm{3,}\mathrm{5,}6\right)$ is used in all stratum populations for any combination of particular choices of $p_{N_l}$ and $p_{{\sigma }_{N_l}},$ $l\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,L.$

An unbiased estimator of the variance of the sample mean requires that $d_l\ge 2,$ for $l\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}L.$ In Neyman and proportional allocations, this assumption may not hold in certain stratum samples when $p_{N_l}$ or $p_{{\sigma }_{N_l}}$ is too small. In this case, we modified the Neyman and proportional allocations to make sure that $d_l\ge 2$ by reducing the maximum $d_l$ and increasing any $d_l$ smaller than 2. These allocation procedures may not be optimal under this modification.

Table 5.1 presents the relative efficiencies of the stratified-PPS-ranked-set sample mean with respect to the stratified-PPS sample mean for the equal, proportional and Neyman allocation procedures. The efficiencies are computed using equations (3.1), (4.1), (4.2) and (4.4). It is clear that the stratified-PPS-ranked-set sample mean has higher efficiency than the stratified-PPS sample mean for all allocation procedures. The efficiency improvement increases with the set size $H.$

Table 5.2 presents the efficiencies of the allocation procedures and the coverage probabilities of the approximate confidence interval for the population mean constructed from the stratified-PPS-ranked-set samples. Again the efficiencies are computed from the analytic expressions in equations (3.1), (4.1), (4.2) and (4.4), but the coverage probabilities are computed from a simulation study by generating 5,000 stratified-PPS-rankek-set samples. The PPS samples are generated using the function ‘lahiri.design’ in the R-package SDaA, Verbeke (2014). Efficiencies of the equal and proportional allocations are compared with respect to the Neyman allocation. Since the Neyman allocation is optimal, we see that all entries, except 0.979 in the last row of column 9, are greater than 1, as expected. The reason that the proportional allocation is better than the Neyman allocation in the last row is that the Neyman allocation is modified. The Neyman allocation yields $d_1\mathrm{<mo>=</mo>}\mathrm{14,}{d}_2\mathrm{<mo>=</mo>}\mathrm{1,}{d}_3\mathrm{<mo>=</mo>}0.$ This allocation is modified to $d_1\mathrm{<mo>=</mo>}\mathrm{11,}{d}_2\mathrm{<mo>=</mo>}\mathrm{2,}{d}_3\mathrm{<mo>=</mo>}2$ so that the cycle size in each stratum sample is greater than 1. The proportional allocation in the last row did not need any modification. Since the Neyman allocation is no longer optimal in this case, it is not as efficient as the proportional allocation.

Neyman allocation is always better than equal allocation even when we modify it for the cycle sizes. The efficiency of proportional allocation with respect to equal allocation can be obtained by dividing column 8 by column 9 in Table 5.2. If the ratio of the entries in column 8 and column 9 is greater than 1, proportional allocation is more efficient than equal allocation.

It is clear that in the first 4 rows of Table 5.2, equal allocation is better than proportional allocation. In these populations, smaller stratum populations have larger variances. Hence, proportional allocation selects less data from the stratum having large variance and more data from the stratum having small variance. In the last four rows of Table 5.2, where large populations have large variances, proportional allocation has higher efficiency than equal allocation since it allocates larger sample sizes to strata with larger variances. These are consistent with the finding in equation (4.5), which indicates that proportional allocation is more efficient when large stratum populations have large variances.

The last three columns of Table 5.2 provide the coverage probabilities of the confidence intervals for the population mean for equal, proportional and Neyman allocation procedures. It is clear that all coverage probabilities are very close to the nominal coverage probability 0.95.

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