Probability-proportional-to-size ranked-set sampling from stratified populations
Section 1. Introduction

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In survey sampling studies, selection of a sampling design depends on the structure of the population. In this paper, we consider a population structure having two main features. It must contain a size variable $X$ and a variable of interest $Y.$ The values of the size variable should be approximately proportional to the values of the $Y$ -variable, and the values of the $X$ -variable should be available for all population units prior to sampling. The second feature of the population structure is that a small percentage of population units should produce extreme values in both the $Y$ - and $X$ -variables with different proportionality constants. These units usually produce larger means and variances than the rest of the units in the population in both the $Y$ - and $X$ -variables. This population structure is very common in practice. In agricultural sampling, a farm population in a state or country may contain two variables, the crop production $Y$ and the farm size $X$ in acres. Farms can be divided into two groups, the farms that have small/normal sizes and the mega-farms that have extremely large values in the $X$ - and $Y$ -variables. The percentage of the mega-farms would be small, but they may have larger means and variances in the $Y$ - and $X$ -variables and the proportionality constant between the $Y$ - and $X$ -values may be larger.

The population structure in the Monthly Retail Trade Survey performed by the United States Census Bureau would be another example. In this case, the population is defined by the business establishments that have Employer Identification Numbers (EINs). The Census Bureau uses a very complex design in which the previous years’ annual revenues are used to construct a size variable. The structure of the population fits the setting we consider in this paper. Revenues from the previous years would be approximately proportional to the current revenues. The revenues for most of the businesses would take typical values, while revenues of a certain percentage of businesses would be extremely large, producing a larger mean and variance and a different proportionality constant. Different proportionality constant may happen because large businesses may be more/less productive than the rest of the businesses.

We provide a third example using apple production data in Turkey in 2002. The data set was collected by the Turkish Statistical Institute and reported in Kadilar and Cingi (2003) and Ozturk and Bayramoglu-Kavlak (2018). It contains two variables, apple production $(Y)$ (in 1,000kg) and the numbers of apple trees $(X)$ in townships (or localities). The sampling units are the 851 localities in the data set. The $X$ -values in all townships are available in the sampling frame prior to sampling. Figure 1.1 provides the scatter plot of the $Y$ - and $X$ -values, where we see that the value of the $Y$-variable is an increasing function of the $X$ -value. We also observe that red-colored points marked with $“X”$ in the plot have large values for both $X$ - and $Y$ -variables and their proportionality constant is different from the other points. Hence, this population fits into our population structure.

The apple production data have additional structures. The entire population is stratified into seven different geographical regions: Marmara, Aegean, Mediterranean, Central Anatolia, Black Sea, Eastern Anatolia and Southeastern Anatolia. These regions have different climate patterns and apple production changes significantly from one region to another. The extreme observations marked with $“X”$ in Figure 1.1 come from the Marmara, Aegean, Mediterranean and Central Anatolia regions. This is a natural setting to construct a PPS-ranked-set sample from each sub-population.

The variable $X$ in our population setting provides information about the relative size of the units in the entire population. Since the variable $Y$ is approximately proportional to the variable $X,$ the size of the unit indicates the importance of its contribution to the variable of interest $Y.$ Hence, important units should have higher probability of being included in the sample. Probability-proportional-to-size (PPS) sampling deliberately imposes higher selection probabilities for important units to produce unbiased and highly efficient estimators for the population mean and/or total. The main contribution of this paper is to introduce a new sampling design which combines the ranking information in a ranked-set sample (RSS) with the advantage of unequal selection probabilities as used in a PPS sample.

A typical PPS sample contains the triplets $\left(Y_i\mathrm{,}{\pi }_i\mathrm{,}{u}_i\right),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}n,$ where $Y_i$ and ${\pi }_i$ are the value of $Y$ and the selection probability of the unit $u_i$ for each draw under sampling with replacement selection. Though PPS sampling can be done without replacement, here all references to PPS sampling refers to sampling with replacement. Readers may refer to Thompson (2002, page 53) for the details of PPS sampling. In a PPS sample, the size variable is not necessarily used directly in the construction of the estimators. On the other hand, the values of the $X$ -variable are available for all population units even for the units not included in the PPS sample. Hence the $X$ -variable could help us to borrow additional information from a comparison set of $H$ unmeasured units.

For the construction of a typical data point $\left(Y_i\mathrm{,}{\pi }_i\mathrm{,}{u}_i\right)$ in a PPS-ranked-set sample, we select $H$ units from the population using PPS sampling with replacement to form a comparison set $\left\{u_1\mathrm{,}\dots \mathrm{,}{u}_H\right\}.$ We rank these units without measurement based on the $X$ -variable with no additional cost. We then measure the value of the $Y$ -variable for only one unit, the unit having rank $R_i,$ obtaining $\left(Y_{\left[R_i\right]}\mathrm{,}{\pi }_{\left[R_i\right]}\mathrm{,}{u}_{\left[R_i\right]}\right),$ where $Y_{\left[R_i\right]}$ and ${\pi }_{\left[R_i\right]}$ are the value of the $Y$ -variable and the selection probability of the unit $u_{\left[R_i\right]}$ at each draw. A data point in the comparison set, $\left(Y_{\left[R_i\right]}\mathrm{,}{\pi }_{\left[R_i\right]}\mathrm{,}{u}_{\left[i\right]}\right),$ provides more information than a data point in a PPS sample, $\left(Y_i\mathrm{,}{\pi }_i\mathrm{,}{u}_i\right),$ since the rank $R_i$ borrows additional information from the other $H-1$ unmeasured units in the comparison set. In this paper, we use this idea to construct a PPS-ranked-set sample that is more informative than a PPS sample. The details of this sampling procedure will be provided in Section 2.

The position information is used in a slightly different context in ranked-set sample (RSS) and judgment-post-stratified (JPS) sampling designs to borrow information from the unmeasured population units. Construction of a ranked-set sample of size $n$ requires one to determine two integers $d$ and $H,$ $n\mathrm{<mo>=</mo>}dH,$ where $H$ and $d$ are the set and cycle sizes, respectively. The set size $H$ controls the amount of information that can be borrowed from the units in comparison sets. The cycle size $d$ is used to increase the total sample size in a RSS. Once $H$ and $d$ are chosen, one then selects $nH$ units from the population and partitions them into $n$ disjoint comparison sets, each having $H$ units. Units in each comparison set are ranked without measurement using the $X$ -variable and the value of the $Y$ -variable $\left(Y_{\left[h\right]j}\mathrm{<mo>;</mo>}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}d\right)$ associated with the $h^{\text{th}}$ ranked $X$ is measured in $d$ different comparison sets, $h\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}H.$ The measured values $Y_{\left[h\right]j}\mathrm{,}h\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}H\mathrm{<mo>;</mo>}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}d,$ are called a ranked-set sample.

The construction of a JPS sample of size $n$ starts with a simple random sample of size $n$ and measures all of them, $Y_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,n.$ For each measured unit in this sample, one then selects an additional $H-1$ units from the population to form a comparison set of size $H.$ The rank $R_i$ of the measured unit $Y_i$ in each of these comparison sets is determined. The pairs of $\left(Y_i\mathrm{,}{R}_i\right),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}n,$ constitute a JPS sample.

The RSS and JPS samples create induced order statistics for the $Y$ -variable through the ranks of the $X$ -variable in comparison sets. Hence, the random variable $Y_{\left[h\right]i}$ $(Y_i$ given that $R_i\mathrm{<mo>=</mo>}h$ for the JPS sample) is stochastically smaller than the random variable $Y_{\left[h^{\prime }\right]j}$ $(Y_i$ given that $R_j\mathrm{<mo>=</mo>}{h}^{\prime })$ for $h\le h^{\prime }.$ This stochastic ordering property induces an implicit stratification among the measured sample units. Efficiency improvements of RSS and JPS samples over a simple random sample can be anticipated from the partition of the total variation into between- and within-strata variation. For further details on these sampling designs, readers may refer to the review paper in Wolfe (2012) and references therein. Both RSS and JPS samples use the position information of the units in the comparison sets, but they do not completely use the information provided by the selection probabilities in a PPS sample. All units in comparison sets for RSS and JPS are selected with equal probabilities. Hence, they may not be appropriate for the population structure that we consider in this paper.

MacEachern, Stasny and Wolfe (2004) introduced the JPS design in an infinite population setting. In a finite population setting, constructions of the JPS and RSS samples depend on whether the comparison sets are selected with or without replacement. Patil, Sinha and Taillie (1995) considered an RSS design in a finite population, where none of the units in a comparison set is returned to the population prior to selection of the next comparison set. Deshpande, Frey and Ozturk (2006) expanded the RSS sampling design with three different without replacement selection policies and constructed nonparametric confidence intervals for population quantiles.

Probability sampling has also generated extensive research interest in RSS and JPS sampling. Al-Saleh and Samawi (2007), Ozdemir and Gokpinar (2007 and 2008), Gokpinar and Ozdemir (2010), Ozturk and Jafari Jozani (2013), Frey (2011) and Ozturk (2014) computed inclusion probabilities and constructed Horwitz-Thompson type estimators for the population mean and total based on a ranked-set sample. These research papers show that an RSS design yields a substantial amount of improvement in efficiency over the usual simple random sampling design. Ozturk (2016) developed estimators for the population mean based on a JPS sample, where he showed that the estimator needs a finite population correction factor similar to the one used in a simple random sample.

A few researchers have applied the RSS methodology to existing survey sampling designs. Muttlak and McDonald (1992) incorporated the RSS sampling design with a line intersect method. Sroka (2008) used it in stratified sampling by constructing an RSS sample from each stratum. Wang, Lim and Stokes (2016) considered the RSS design in a cluster randomized design with a mixed effect model, where the cluster effect is treated as random. They showed that use of RSS at the cluster level has much bigger impact on efficiency than using the RSS at the within-cluster level. Nematollahi, Salehi and Aliakbari Saba (2008) used the RSS design in a finite population setting only in the second stage of a two-stage sampling with replacement selection scheme. Since they use the RSS design only in the second stage with replacement, the efficiency improvement of their estimator with respect to a two-stage SRS sample estimator was minimal. Sud and Mishra (2006) also used a two-stage cluster sample with ranked set sampling design in a finite population setting under the assumption that the cluster population sizes are all equal. Ozturk (2019a) developed design based statistical inference for a two-stage clustered ranked-set sample in a finite population setting.

In this paper, we develop statistical inference for the PPS-ranked-set sampling design in a population setting where the values of the size variable are roughly proportional to the values of the variable of interest and a small percentage of population units produces large $X$ - and $Y$ -values with a different proportionality constant. We motivate the new sampling design using apple production data. Section 2 introduces the PPS-ranked-set sample in a finite population setting. It constructs unbiased estimators for the population mean, total and their variances. We show that the PPS-ranked-set sample estimator has smaller variance than a PPS sample estimator. Section 3 extends the PPS-ranked-set sample to a stratified population and constructs unbiased estimators for the population mean, total and their variances. Section 4 considers four different sample size allocation procedures to minimize the variance of the estimator under a cost model and different stratum population structures. Section 5 provides an efficiency comparison for the PPS-ranked-set sample estimator of the population mean with respect to other competing estimators. Section 6 illustrates the use of PPS-ranked-set sample data to estimate the apple production in Turkey. Section 7 provides some concluding remarks. All proofs are given in the Appendix.

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