Model based inference using ranked set samples
Section 4. Cost model

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Efficiency improvement of the RSS estimator results from the relative position (rank) information of the measured observation among unmeasured $H-1$ units in a set. This extra information comes at the cost of sampling a set of size $H$ and obtaining the subsequent ranking. Ranking can be performed either using concomitant (auxiliary) variable or visual inspection of the physical units in each set. Hence, these two approaches, visual and concomitant ranking models, may lead to different cost structures. In either case, there needs to be some sort of consistency in ranking process to develop a meaningful cost function. Patil, Sinha and Taillie (1997) defined a coherent ranking process in which ranking of a set is consistent for all subsets and supersets. Under a coherent ranking scheme, the rank order of $H$ units would remain identical when ranking any of their subsets or supersets containing them. For further detail in coherent ranking, readers are referred to Patil et al. (1997) or Nahhas, Wolfe and Chen (2002).

Concomitant ranking uses an auxiliary variable to rank $H$ units in a set. The quality of ranking depends on monotonic (not necessarily to be linear) relationship between the variable of interest and auxiliary variable. On the other hand, visual inspection can be performed in different ways. One of the strategy is to use pairwise comparison. Under coherent ranking, not all $\left(\begin{array}{c}H\\ 2\end{array}\right)$ pairwise comparisons are necessary for a visual ranking. For example, in a set of size $H\mathrm{<mo>=</mo>\; 3},$ if unit 1 is judged to be smaller than unit 2 and unit 2 is smaller than unit 3, we reasonably assume unit 1 is less than unit 3 without a comparison. In order to differentiate the impact of the cost structures of the concomitant and visual ranking schemes, we denote the estimator in equation (2.3) with ${\overline{Y}}_{\text{RC}}$ for concomitant ranking and ${\overline{Y}}_{\text{RV}}$ for visual ranking.

For visual ranking, we use visual inspection model of Nahhas et al. (2002). This model always compares the selected unit with the largest element previously ranked. It chooses a unit at random and compares it with the unit previously judged to be largest. If it is judged to be larger, then it becomes the largest among all judged units. Otherwise, it is compared with the next largest previously judged unit until it is assigned a rank. The number of required pairwise comparisons under this ranking strategy with a coherent ranking scheme is an integer valued random variable having the support $H-1,H,H+1,\dots ,\left(\begin{array}{c}H\\ 2\end{array}\right).$ The expected number of pairwise comparison for this ranking scheme is approximately equal to $f\left(H\right)\mathrm{<mo>=</mo>}\left(H+2\right)\left(H-1\right)/4.$ The reader is referred to Nahhas et al. (2002) for further development on expected number of pairwise comparisons.

We now introduce cost definitions for three models; concomitant, visual ranking and simple random sampling models: $C_C\mathrm{<mo>=</mo>}$ total cost for concomitant ranking, $C_V\mathrm{<mo>=</mo>}$ total cost for visual ranking, $C_S\mathrm{<mo>=</mo>}$ total cost for simple random sampling, $c_i\mathrm{<mo>=</mo>}$ cost of sampling a single unit, $c_{qy}\mathrm{<mo>=</mo>}$ cost of quantification of the variable of interest $\left(Y\right)$ for one unit, $c_{qx}\mathrm{<mo>=</mo>}$ cost of quantification of concomitant (auxiliary $X)$ variable for one unit, $c_r\mathrm{<mo>=</mo>}$ cost of one pairwise comparison. We assume that overhead cost in SRS model to be zero, but the overhead cost (in excess of the overhead cost of SRS) of RSS concomitant (visual) ranking model is absorbed in $c_{qx}\left(c_r\right).$ Total cost for these three models are then given by

$$C_S\mathrm{<mo>=</mo>}{n}_s\left(c_i+c_{qy}\right)\mathrm{,}{C}_C\mathrm{<mo>=</mo>}{n}_c\left(H{c}_i+H{c}_{qx}+c_{qy}\right)\mathrm{,}{C}_V\mathrm{<mo>=</mo>}{n}_V\left(H{c}_i+f\left(H\right)+c_{qy}\right)\mathrm{,}$$

where $n_S\text{},n_C$ and $n_V$ are the total (measured) observations in SRS, RSS concomitant and RSS visual ranking models. Readers are referred to Nahhas et al. (2002) for further details on these cost functions.

We now fix the total cost on these three models $C_S\mathrm{<mo>=</mo>}{C}_C\mathrm{<mo>=</mo>}{C}_V\mathrm{<mo>=</mo>}C.$ Under this fixed cost, we look at the relative efficiency of ${\overline{Y}}_{\text{RC}}$ and ${\overline{Y}}_{\text{RV}}$ with respect to SRS sample mean ${\overline{Y}}_{\text{SRS}}.$ Let

$$\text{RP}\mathrm{<mo>=</mo>}\frac{1}{1-D}\mathrm{,}D\mathrm{<mo>=</mo>}1-\frac{1}{H{\sigma }^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2\text{}\mathrm{,}$$

where RP is the relative precision of RSS sample mean with respect to SRS sample mean in an infinite population setting. Under super population model, we can establish the following efficiency result.

Theorem 3 Let $Y_{\left[h\right]i}\text{},$   $h\mathrm{<mo>=</mo>\; 1,}\dots ,H,$   $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}d,$  be a ranked set sample from a finite population $P^N\text{}.$  For a fixed cost, under super population model and coherent ranking scheme, the following efficiency results are established.

$$\begin{array}{lll}\text{RE}\left({\overline{Y}}_{\text{RC}}\text{}\mathrm{,}{\overline{Y}}_{\text{SRS}}\right)\hfill & \mathrm{<mo>=</mo>}\frac{\text{Var}\left({\overline{Y}}_{\text{SRS}}\right)}{\text{Var}\left({\overline{Y}}_{\text{RC}}\right)}\ge \mathrm{1,}\hfill & \text{if}\text{RP}\ge \frac{H{c}_i+H{c}_{qx}+c_{qy}}{c_i+c_{qy}}\hfill \\ \text{RE}\left({\overline{Y}}_{\text{RV}}\text{}\mathrm{,}{\overline{Y}}_{\text{SRS}}\right)\hfill & \mathrm{<mo>=</mo>}\frac{\text{Var}\left({\overline{Y}}_{\text{SRS}}\right)}{\text{Var}\left({\overline{Y}}_{\text{RV}}\right)}\ge \mathrm{1,}\hfill & \text{if}\text{RP}\ge \frac{H{c}_i+f\left(H\right)c_r+c_{qy}}{c_i+c_{qy}}\mathrm{.}\hfill \end{array}$$

The fractions on the right hand side of the inequalities in the above theorem is the ratio of the cost of selecting and measuring a single unit in RSS and SRS, respectively. If the cost of sampling a unit and cost of ranking a set are negligible (free), the cost ratio becomes 1. One of the basic assumptions, in settings where RSS is used, is that ranking cost of units is relatively cheap with respect to the cost of measurement. Hence, it is not unreasonable to assume that cost ratio will be very close to 1 for settings where use of RSS is appropriate. It is established in the literature that RP is always greater than or equal to 1 (see Dell and Clutter (1972), Patil et al. (1997), Nahhas et al. (2002)). It is equal to 1 only under random ranking. The values of RP for normal population for different values of $\rho$ (correlation coefficient between response $Y$ and auxiliary variable $X)$ and set sizes are given in Table 4.1. It is now reasonable to say that RSS estimator under super population model is more efficient if the cost of sampling and ranking a unit is relatively cheap in comparison with measurement cost.

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