Model based inference using ranked set samples
Section 1. Introduction

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In many survey sampling studies, it is very common that the sampling frame has additional auxiliary information in addition to characteristic of interest. Under a fairly strong modeling assumption, this auxiliary information improves the statistical inference. For example, ratio and regression estimators use covariate information under a linearity assumption to estimate the population mean or total. The auxiliary information can also be used under a weaker assumption in a ranked set sample (RSS) and judgment post stratified (JPS) sample. These samples use auxiliary information to increase the information content of each measured unit through a ranking process. The ranking process is performed in a small set of size $H$ formed by combining the measured unit with an additional $H-1$ unmeasured units from the population. Ranking process is performed either before or after measurement and determines the relative position of each measured unit. Ranking information can be obtained from either a visual inspection or some other form of ranking process. A reasonable ranking mechanism requires some sort of monotonic relationship between the ranking variable and response, which is much weaker than the strong linearity assumption of regression and ratio estimators.

A balanced ranked set sample of set size $H$ and cycle size $d$ can be constructed by first selecting $n\mathrm{<mo>=</mo>}Hd$ simple random samples of size $H$ from the population and ranking the units in each sample without measurement from smallest to largest. In these $n$ ranked sets (samples), one then measures the units with rank 1 in the first $d$ sets, the unit with rank 2 in the next $d$ sets and so on. This yields samples of $H$ different sets of judgment order statistics, each of which has $d$ independent and identically distributed judgment order statistics.

A sharp contrast exists between an observation from SRS and RSS, where the observation from an SRS sample provides information only about the unit on which it was measured while the observation from an RSS sample, in addition to the information that the measured unit provides, also provides limited information about the other $\left(H-1\right)$ unmeasured units in the set through the relative position (rank) of measured unit. Since ranking process does not require a formal measurement and is usually less expensive in comparison with formal measurement, the RSS sample provides substantial amount of reduction in sampling cost.

A JPS sample differs from an RSS sample in that the ranking step comes after the construction of an SRS sample. Construction of a JPS sample of size $n$ requires a set size $H.$ Once the set size $H$ is determined, one first draws a simple random sample of size $n$ and makes a measurement on each of the $n$ units. For each measured unit in the sample, one then selects additional $H-1$ units to form a set of size $H.$ The units in this set are ranked from smallest to largest without measurement and the rank of the measured unit in the set is recorded. The JPS sample then consists of $n$ measured values, together with their ranks.

Both RSS and JPS samples induces a stochastic structure among measured units in which observations in judgment class $h$ are usually smaller than the observations in judgment class $h^{\prime }\text{},$ $h\mathrm{<mo><</mo>}{h}^{\prime }\text{}.$ This stochastic ordering feature spreads the measured units in the support of the distribution and creates a better representative sample than a simple random sample. The nature of stochastic ordering in a JPS sample is significantly different from the stochastic ordering in an RSS sample. A JPS sample consists of a simple random sample and an associated rank vector. This rank vector is loosely related to the sample and may be ignored if desired. On the other hand, an RSS sample is measured as judgment order statistics, judgment ranks can not be separated from the observed values. An RSS sample can not be treated as an SRS sample.

Both JPS and RSS sampling designs have generated extensive research interest in a finite population setting. Patil, Sinha and Taillie (1995) used ranked set sample to estimate population mean for a population of size $N$ when the sample is constructed without replacement. Takahasi and Futatsuya (1998) showed that the ranked set sample estimator of the population mean is more precise than the simple random sample estimator when samples are drawn without replacement from a finite population. Deshpande, Frey and Ozturk (2006) described three different sampling designs and constructed nonparametric confidence intervals for population quantiles. Al-Saleh and Samawi (2007), Ozdemir and Gokpinar (2007 and 2008), Gokpinar and Ozdemir (2010), Ozturk and Jozani (2013), Frey (2011) and Ozturk (2014, 2015, 2016a) computed inclusion probabilities and constructed Horwitz-Thompson type estimators for 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 (2016b) developed estimators for 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.

All available research in literature in JPS and RSS sampling designs in a finite population setting considers design-based approach. To our knowledge, super population model has not been used. In this paper, we develop a model-based statistical inference using RSS sampling design for population mean and total in a finite population setting. Similar results, with some additional variation due to random judgment class samples sizes, can also be established for a JPS sampling design. Because of the random judgment class sample sizes, the estimators based on a JPS sample are less efficient than the estimators based on an RSS sample. For this reason, the JPS sample is not considered further in this paper. Section 2 clearly defines the model and describes the sampling designs for RSS under super population model. We show that estimators of population mean and total are model-unbiased and their mean square prediction errors (MSPE) are smaller than the MSPE of the same estimators of an SRS sample. Section 3 constructs unbiased estimators for the MSPE and provides approximate confidence intervals for the population mean and total. Section 4 introduces cost models to account the effect of additional cost (excess of the cost of construction of SRS sample) in construction of RSS sample. Section 5 provides empirical evidence about the performance of the estimators. Section 6 applies the proposed estimators to an example in a finite population setting. Section 7 provides some concluding remarks.

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