Statistical inference based on judgment post-stratified samples in finite population Section 4. Empirical resultsStatistical inference based on judgment post-stratified samples in finite population Section 4. Empirical results

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In this section, we look at the finite sample properties of the estimators in a small scale simulation study under wide ranges of simulation parameters. Data sets are generated from discrete normal and discrete shifted exponential populations for given population size $N.$ The discrete populations are constructed from the quantile function

$$x_i\mathrm{<mo>=</mo>}{F}^{-1}\left(\frac{i}{N+1}\right)\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}N\mathrm{,}(4.1)$$

where $F$ is either normal or exponential cumulative distribution functions (CDF). For discrete normal population, we used location parameter 10 and scale parameter 4. For shifted discrete exponential population, we use the CDF of standard exponential distribution to generate $x_i$ in equation (4.1) and then shift each $x_i$ by adding 10. The population size is taken to be $N\mathrm{<mo>=</mo>\; 150.}$

We used sample $\left(n\right)$ and set size $\left(H\right)$ to have integer values for $n/H$ so that a balanced ranked set sample of size $n$ can be created. Sample and set size combinations $\left(n\mathrm{,}H\right)$ are $\left(\mathrm{10,}2\right),$ $\left(\mathrm{15,}3\right),$ $\left(\mathrm{20,}4\right),$ $\left(\mathrm{25,}5\right).$ To control the quality of ranking information we used auxiliary variable $Y,$ where $\rho \mathrm{<mo>=</mo>}\text{cor}\left(X\mathrm{,}Y\right)$ with $\rho \mathrm{<mo>=</mo>\; 1,}\mathrm{0.75.}$ The value of $\rho \mathrm{<mo>=</mo>\; 1}$ yields perfect ranking and the value of $\rho \mathrm{<mo>=</mo>\; 0.75}$ creates errors in ranking. Simulation size is taken to be 3,000. Rao-Blackwellized estimators are computed from Algorithm 1 with $B\mathrm{<mo>=</mo>\; 50}$ and bootstrap replication size 200.

The first part of the simulation investigates the efficiencies of the estimators and coverage probability of the confidence intervals of the population mean. All estimators are compared with design-2 Rao-Blackwellized estimators $\left({\tilde{\mu }}_2\right).$ Let $D\left(X\right)$ be any one of the estimators introduced in Section 2 and 3. The relative efficiency of $D\left(\mathrm{.}\right)$ with respect to ${\tilde{\mu }}_2$ is given by

$$R\left(D\right)\mathrm{<mo>=</mo>}\frac{MSE\left(D\right)}{MSE\left({\tilde{\mu }}_2\right)}\mathrm{,}(4.2)$$

where $MSE\left(D\right)$ is the estimated mean square error of estimator $D.$ In equation (4.2), the value $R\left(D\right)\mathrm{<mo>></mo>\; 1}$ indicates that the estimator ${\tilde{\mu }}_2$ is more efficient than the estimator $D.$

We consider two types of confidence intervals for the population mean. Percentile confidence interval based on bootstrap distribution is given in Section 3. The coverage probabilities of these intervals will be labeled with $C^a\left({\tilde{\mu }}_0\right)$ for design-0 and $C^a\left({\tilde{\mu }}_2\right)$ for design-2. A second type of an approximate confidence interval can be constructed from standard theory. Note that we have unbiased estimators, ${\widehat{\sigma }}_r,$ for the variances of ${\widehat{\mu }}_r;$ $r\mathrm{<mo>=</mo>\; 0,}2.$ A $100\left(1-\gamma \right)\mathrm{<mi>\%</mi>}$ confidence interval for $\mu$ is then given by

$${\widehat{\mu }}_r\pm t_{n-\mathrm{1,1}-\gamma /2}{\widehat{\sigma }}_r\mathrm{<mo>;</mo>}r\mathrm{<mo>=</mo>\; 0,}\mathrm{2,}$$

where $t_{n-\mathrm{1,1}-a}$ is the $a^{\text{th}}$ upper quantile of the t-distribution with $n-1$ degrees of freedom. The coverage probabilities of these confidence intervals will be labeled as $C^b\left({\widehat{\mu }}_0\right)$ for design-0 and $C^b\left({\widehat{\mu }}_2\right)$ for design-2. Ahn, Lim and Wang (2014) suggested using $n-H$ degrees of freedom for the $\mathrm{t-}$approximation. This selection may also work in JPS sampling in finite population setting with some increased variation due to unbalanced nature of a JPS sample. This line of work, on the other hand, is not persuaded in this paper because of the space limitation.

Table 4.1 presents the relative efficiencies of the estimators and the coverage probabilities of the confidence intervals for discrete normal populations. It is clear that Rao-Blackwellized design-2 estimator $\left({\tilde{\mu }}_r\right)$ outperforms all the other estimators including RSS estimators. In general RSS estimators are more efficient than JPS estimators due to random judgment class sample size vector $M.$ This can be seen in Table 4.1 by looking at the ratio

$$\frac{R\left({\widehat{\mu }}_r\right)}{R\left({\mu }_r^{\mathrm{*}}\right)}\mathrm{<mo>=</mo>}\frac{MSE\left({\widehat{\mu }}_r\right)}{MSE\left({\mu }_r^{\mathrm{*}}\right)}\mathrm{,}r\mathrm{<mo>=</mo>\; 0,}2.$$

For $r\mathrm{<mo>=</mo>\; 0},$ $\rho \mathrm{<mo>=</mo>\; 1}$ and sample-set size combinations $\left(n\mathrm{,}H\right),$ $\left(\mathrm{10,}2\right)\mathrm{,}\left(\mathrm{15,}3\right)\mathrm{,}\left(\mathrm{20,}4\right)\mathrm{,}\left(\mathrm{25,}5\right),$ these ratios are $1.267\left(1.698/1.340\right),$ $1.491\left(2.117/1.419\right),$ $1.815\left(2.985/1.644\right),$ $2.391\left(3.479/1.455\right),$ respectively. It is obvious that ranked set sample estimator ${\mu }_0^{\mathrm{*}}$ is more efficient that JPS estimator ${\widehat{\mu }}_0.$ This can be explained from the fact that RSS sample uses a constant (nonrandom) sample size vector $m\mathrm{<mo>=</mo>}\left(n_1\mathrm{,}\dots \mathrm{,}{n}_H\right).$ Hence there is not extra variation due to randomness of $M$ in JPS sample and this yields smaller variance for the estimator.

Table 4.1 (entries in columns $R\left({\mu }_0^{\mathrm{*}}\right)$ and $R\left({\mu }_2^{\mathrm{*}}\right))$ indicates that Rao-Blackwellized JPS estimators are better than RSS estimators. In this case, there is a clear difference between Rao-Blackwellized JPS estimators and RSS sample estimators. In RSS sample, even though $m$ is constant, ranking information (or rank $R_i$ that belongs to each $X_i)$ is obtained from a particular construction of $n$ sets, each of size $H.$ On the other hand, Rao-Blackwellized JPS estimators consider all possible constructions of $n$ sets, each of size $H.$ Hence, the content of ranking information is richer in a Rao-Blackwellized JPS sample than the content of ranking information of an RSS sample. This increased ranking information makes Rao-Blackwellized estimators superior to RSS estimators.

Table 4.1 also presents coverage probabilities of the confidence intervals. The coverage probabilities of bootstrap percentile confidence intervals are slightly lower than the nominal value 0.95. The coverage probabilities of the confidence intervals based on $\mathrm{t-}$distribution are reasonably close to nominal coverage probability 0.95.

Table 4.2 provides variance estimates of the mean estimators from simulation and the estimators in equations (2.5), (2.6), (2.8), and (3.2) in Sections 2 and 3. We already proved that the estimators ${\widehat{\sigma }}_{{\widehat{\mu }}_r}^2,$ $r\mathrm{<mo>=</mo>\; 0,}2,$ are unbiased. Entries for these variance estimators are very close to the corresponding values based on simulated variance estimates. The truncated variance estimator is almost identical to the un-truncated unbiased estimator. This shows that negative values happen rarely and there is not much difference between the truncated and un-truncated variance estimators. The bootstrap variance estimates of Rao-Blackwellized estimators are also very close to simulated variance estimates. Patterns similar to the ones we observed in Tables 4.1 and 4.2 also hold in Tables 4.3 and 4.4 for shifted exponential population.

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