Statistical inference based on judgment post-stratified samples in finite population Section 1. IntroductionStatistical inference based on judgment post-stratified samples in finite population Section 1. Introduction

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In many survey sampling studies, in addition to variable of interest, sampling frame has additional available auxiliary variables to improve the information content of a sample. These auxiliary variables have been successfully used to construct better estimators, such as ratio and regression estimators. These estimators usually require strong modeling assumptions between the auxiliary variable(s) and variable of interest. MacEachern, Stasny and Wolfe (2004) introduced judgment post-stratified (JPS) sample, and constructed estimators that require weaker modeling assumptions than the ratio and regression estimators.

A JPS sample selects a simple random sample of size $n$ from a population and measures all selected units, $X_i;$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$ For each one of the measured unit, researcher selects additional $H-1$ units to form a set of size $H.$ This set contains the measured unit $X_i$ and the additionally selected $H-1$ units. Units in these sets are ranked from smallest to largest without a measurement and the rank of $X_i$ is determined. The pairs $\left(X_i\mathrm{,}{R}_i\right)\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ are called a JPS sample. Ranking process in these sets can be performed either using visual inspection of the units or some available auxiliary variable. If the visual inspection is used, rankers should be blinded to actual values of $X_i$ to avoid any bias. If the auxiliary variable is used, a monotonic relationship between the variable $X$ and auxiliary variable is required. These assumptions are much weaker than the linearity assumption in regression and ratio estimators.

Ranking information in a JPS sample is used to induce a structure among measured observations by creating $H$ judgment classes of similar units. The judgment class $h,$ $h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H,$ contains all measured observations with judgment rank $h.$ Since rank $R_i$ provides information about the relative position of $X_i$ among $H$ units in a set, observations in judgment class $h$ are stochastically larger than the observations in judgment class $h^{\prime },$ for $h^{\prime }\mathrm{<mo><</mo>}h.$ This induced structure increases the information content of the sample. One may also view a JPS sample as a stratified sample with $H$ strata. In this case, the improved efficiency can be established from standard theory of stratified sampling in survey sampling designs.

In a JPS sample, ranks are determined after a simple random sample is selected. Thus, the number of observations, $M_h,$ in judgment class $h$ is a random variable. The joint distribution of $M\mathrm{<mo>=</mo>}\left(M_1\mathrm{,}\dots \mathrm{,}{M}_H\right)$ is multinomial with parameters $n$ and success probability vector $\left(1/H,\dots ,1/H\right).$ Since $M$ is a random variable, it is highly possible that $M_h\mathrm{<mo>=</mo>\; 0}$ for some $h$ when the sample size $n$ is small. Statistical inference then should account for the impact of empty strata on the procedures.

In an infinite population setting, JPS sample has generated extensive research interests. For a tiny slice of literature, readers are referred to Frey and Feeman (2012, 2013), Frey and Ozturk (2011), Stokes, Wang and Chen (2007), Wang, Lim and Stokes (2008), Wang, Stokes, Lim and Chen (2006), Wang, Wang and Lim (2012), Ozturk (2013, 2014a, 2014b, 2015) and the references there in.

One way to avoid having random sample size $M_h$ is to rank the units in each set before selecting a simple random sample from the population. In this case sampling design is called ranked set sample (RSS). Ranked set sampling is introduced in McIntyre (1952, 2005) to estimate the population mean in agricultural research. To construct an RSS sample of size $n,$ researcher first determines the design parameters, set size $H$ and the judgment class sample size vector $m\mathrm{<mo>=</mo>}\left(m_1\mathrm{,}\dots \mathrm{,}{m}_H\right),$ where $m_h$ is the required number of observations to be selected in judgment class $h.$ Researcher next selects $nH$ units at random from the population and divide them into $n$ sets, each of size $H.$ Units in each one of these sets are ranked and the $h^{\text{th}}$ judgment order statistics is measured in $m_h$ sets so that ${\displaystyle {\sum }_{h\mathrm{<mo>=</mo>\; 1}}^H}{m}_h\mathrm{<mo>=</mo>}n.$ The measured observations $X_{\left[h\right]j};$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}{m}_h;$ $h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H$ are called an unbalanced ranked set sample, where $X_{\left[h\right]j}$ is judgment order statistics from a set of size $H.$ If the judgment class sample sizes are all equal $m_h\equiv n/H;$ $h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H,$ the sample is called a balanced ranked set sample. If there is no ranking error, judgment order statistics become usual order statistics from a sample of size $H.$ In this case, usual order statistic notation is used to denote the $h^{\text{th}}$ order statistic, $X_{\left(h\right)j}.$

In recent years, there have been increased research activities in JPS and RSS sampling 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. Deshpande, Frey and Ozturk (2006) expanded the without replacement policy in Patil et al. (1995) into three different designs, design-0, design-1 and design-2, and constructed confidence intervals for population quantiles. The design-0 constructs the sample by replacing all units back into the population prior to selection of the next set. Design-1 constructs the sample by replacing only the unmeasured units back into the population before selecting the next set. Design-2 constructs the sample by replacing none of the units back into population regardless of whether they were measured or not. Al-Saleh and Samawi (2007), Ozdemir and Gokpinar (2007 and 2008), Jafari Jozani and Johnson (2011, 2012), Gokpinar and Ozdemir (2010), Ozturk and Jafari Jozani (2013), and Frey (2011) computed inclusion probabilities and constructed Horvitz-Thompson type estimators for population mean and total for some variant of design-0, design-1 and design-2 samples based on ranked set samples.

Ozturk (2014a) combined ranking information from different sources in a ranked set sample, estimated the inclusion probabilities of population units and constructed estimator for population mean. For settings where population values of auxiliary variables are available, Ozturk (2016) used population ranks (global ranking information) of selected sample units to induce stronger structure in data to improve the information content of the sample. He showed that samples constructed based on global ranking information provides higher efficiencies. A comprehensive up to date literature review both in JPS and RSS can be found in recent review paper in Wolfe (2012).

In this paper, we consider with- and without-replacement sampling designs for JPS sampling in finite population setting. Section 2 provides detailed descriptions for the construction of the designs. For each design, we obtain the probability mass functions, means, variances and covariances of order statistics. These results are used to construct unbiased estimator for the population mean and unbiased estimators for the variance of sample means. Section 3 constructs Rao-Blackwellized estimators by conditioning on the measured observations $X_i;$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$ Section 4 provides empirical evidence for the new estimators. Section 5 applies the proposed procedures to 2012 United States Agricultural Census (USDA) data. Section 6 provides some concluding remarks. The proofs of the theorems are provided in Appendix.

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