Statistical inference based on judgment post-stratified samples in finite population Section 6. Concluding remarkStatistical inference based on judgment post-stratified samples in finite population Section 6. Concluding remark

We have developed two sampling designs for judgment post stratified samples in a finite population setting. The designs are constructed with two levels of without-replacement policies, design-0 and design-2. Design-0 is constructed by replacing all measured and unmeasured units in a set back in population before selecting the next units. Design-2 is constructed by using without-replacement policy on all units regardless of the measurement status. Hence, random variables in design-0 are independent, but random variables in design-2 are negatively correlated. In these designs, measured observations are ranked and stratified into $H$ judgment classes after a simple random sample is collected. Using this ranking information, we construct unbiased estimators for the population mean and the variance of these unbiased estimators of population mean. We showed that the new estimators based on level-2 design outperform simple random sample mean, design-0 JPS sample mean and ranked set sample mean estimators. Main focus of this paper was on the estimation of populations mean, but the results also apply to estimation of population total with a minor adjustment in notation.

Post stratification creates random ranks and random number of observations in judgment classes. By conditioning on the measured values in the sample, we construct Rao-Blackwellized estimators by computing the conditional expected value of the estimators over all possible values of random ranks. Rao-Blackwellized estimators are unbiased and more efficient than unconditional estimators. We construct finite sample bootstrap inference for the population mean based on all proposed estimators. The new sampling designs and estimator are applied to 2012 USDA census data to show that they are viable sampling designs and estimators in survey sampling studies.

In one of our current projects, we extend these design-0 and design-2 to two-stage sampling where primary and secondary sampling units constitute two finite populations. In this case, we expect that some interesting optimization problem will arise related the selection of sample sizes and design-0 and design-2 in stage I and II sampling.

Appendix

Proof of Lemma 1: The proof of part (i) is given in Presnell and Bohn (1999) in an infinite population setting. The proof is essentially the same in finite population setting.

For the proof of part (ii), we consider the joint probability mass function of $X_{j}$ and $X_{t}$ given their judgment ranks $R_{j} = h$ and $R_{t} = h^{'}$

$P (X_{j} = x, X_{t} = y | R_{j} = h, R_{t} = h^{'}) = P (X_{[h]} = x, X_{[h^{'}]} = y); x \neq y, (x, y) \in P .$

For $h =1, \dots, H,$ let $J_{h} = i$ be the event that $i^{th}$ order statistic in a set of size $H$ is judged to be the $h^{th}$ judgment order statistics. Using this notation, we write

$P (X_{[h]} = x, X_{[h^{'}]} = y) = \sum_{i =1}^{H} \sum_{k =1}^{H} P (X_{(i)} = x, X_{(k)} = y, J_{h} = i, J_{h^{'}} = k); x \neq y, (x, y) \in P . (A .1)$

For each $i,$ the events ${J_{h} = i}; h =1, \dots, H,$ partition the sample space of random variable $R_{j}$ because the $i^{th}$ order statistic is assigned to one and only one judgment rank. Since rank $R_{t}$ is obtained independently from another disjoint set using the same ranking procedure, the events ${J_{h^{'}} = k}; h^{'} =1, \dots, H,$ also partition the sample space. Hence, we write

$\sum_{h =1}^{H} \sum_{h^{'} =1}^{H} P (X_{(i)} = x, X_{(k)} = y, J_{h} = i, J_{h^{'}} = k) = P (X_{(i)} = x, X_{(k)} = y); x \neq y, (x, y) \in P . (A .2)$

Combining equations (A.1) and (A.2), we obtain

$\begin{array}{l} \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} P (X_{[h]} = x, X_{[h^{'}]} = y) & = \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} \sum_{i =1}^{H} \sum_{k =1}^{H} P (X_{(i)} = x, X_{(k)} = y; J_{h} = i, J_{h^{'}} = k) \\ = \sum_{i =1}^{H} \sum_{k =1}^{H} P (X_{(i)} = x, X_{(k)} = y); x \neq y, (x, y) \in P . \end{array}$

This completes the proof.

Proof of Theorem 1:

(i) We use the conditional expectation to write

$E ({\hat{μ}}_{r}) = \sum_{h =1}^{H} E (\frac{I_{h}}{d_{n} M_{h}}) \sum_{i =1}^{n} E (X_{i} I (R_{i} = h) | R_{i} = h) = \sum_{h =1}^{H} E (\frac{I_{h} M_{h}}{d_{n} M_{h}}) E X_{[h]} .$

Random variables $I_{h} / d_{n};$ $h =1, \dots, H,$ are identically distributed. By using part (i) in Lemma 3, the above expectation can be written

$E ({\hat{μ}}_{r}) = E (\frac{I_{1}}{d_{n}}) \sum_{h =1}^{n} E X_{[h]} = \frac{1}{H} \sum_{h =1}^{H} μ_{[h]} = \frac{1}{H} \sum_{h =1}^{H} μ_{(h)} = μ, r =0, 2$

which completes the proof. The last two equalities in the above equation follows from part (i) of Lemma 1.

(ii) For the the proof of $Var ({\hat{μ}}_{0}),$ we use conditional variance

$Var ({\hat{μ}}_{0}) = E {Var ({\hat{μ}}_{0} | R)} + Var {E ({\hat{μ}}_{0} | R)} .$

Since $X_{j}$ are selected with replacement, they are all independent. Hence, the first term in the above equation yields

$E {Var ({\hat{μ}}_{0} | R)} = \sum_{h =1}^{H} E (\frac{I_{h}^{2} M_{h}}{d_{n}^{2} M_{h}^{2}}) σ_{[h]}^{2} = E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h]}^{2} .$

We now consider

$\begin{array}{l} Var {E ({\hat{μ}}_{0} | R)} & = \sum_{h =1}^{H} Var (\frac{I_{h}}{d_{n}}) μ_{[h]}^{2} + \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} Cov (\frac{I_{h}}{d_{n}}, \frac{I_{h^{'}}}{d_{n}}) μ_{[h]} μ_{[h^{'}]} \\ = Var (\frac{I_{1}}{d_{n}}) \sum_{h =1}^{H} μ_{[h]}^{2} + Cov (\frac{I_{1}}{d_{n}}, \frac{I_{2}}{d_{n}}) \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} μ_{[h]} μ_{[h^{'}]} \\ = \frac{H}{H - 1} Var (\frac{I_{1}}{d_{n}}) \sum_{h =1}^{H} {(μ_{[h]} - μ)}^{2} \end{array}$

which completes the proof of $Var ({\hat{μ}}_{0}) .$ The last equality is obtained by using part (iv) of Lemma 3.

For the proof of $Var ({\hat{μ}}_{2}),$ we again consider the conditional variance of ${\hat{μ}}_{2}$ given the ranks

$Var ({\hat{μ}}_{2}) = E (Var ({\hat{μ}}_{2} | R)) + Var (E ({\hat{μ}}_{2} | R)) .$

The second term in the right hand side of the above equation can be written

$\begin{array}{l} Var (E ({\hat{μ}}_{2} | R)) & = Var (\sum_{h =1}^{h} \frac{μ_{[h]} I_{h}}{d_{n}}) \\ = \sum_{h =1}^{H} Var (\frac{I_{h}}{d_{n}}) μ_{[h]}^{2} + \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} Cov (\frac{I_{h}}{d_{n}}, \frac{I_{h^{'}}}{d_{n}}) μ_{[h]} μ_{[h^{'}]} \\ = Var (\frac{I_{1}}{d_{n}}) \sum_{h =1}^{H} μ_{[h]}^{2} + Cov (\frac{I_{1}}{d_{n}}, \frac{I_{2}}{d_{n}}) \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} μ_{[h]} μ_{[h^{'}]} . \end{array}$

Note that $Cov (I_{1} / d_{n}, I_{2} / d_{n}) = - Var (I_{1} / d_{n}) / (H - 1)$ in part (iv) of Lemma 3. Using this equality in the above equation, we obtain

$Var (E ({\hat{μ}}_{2} | R)) = \frac{H}{H - 1} Var (\frac{I_{1}}{d_{n}}) \sum_{h =1}^{H} {(μ_{[h]} - μ)}^{2} . (A .3)$

We now consider $Var ({\hat{μ}}_{2} | R)$

$\begin{array}{l} Var ({\hat{μ}}_{2} | R) & = Var {\sum_{h =1}^{H} \frac{I_{h}}{d_{n} M_{h}} \sum_{j =1}^{n} X_{j} I (R_{j} = h) | R} \\ = \sum_{h =1}^{H} \frac{I_{h}^{2}}{d_{n}^{2} M_{h}^{2}} (M_{h} σ_{[h]}^{2} + M_{h} (M_{h} - 1) σ_{[h, h]}) + \sum_{h}^{H} \sum_{h^{'} \neq h}^{H} \frac{I_{h} I_{h^{'}}}{d_{n}^{2} M_{h} M_{h^{'}}} M_{h} M_{h^{'}} σ_{[h, h^{'}]} \\ = \sum_{h =1}^{H} \frac{I_{h}^{2}}{d_{n}^{2} M_{h}} σ_{[h]}^{2} + \sum_{h =1}^{H} \frac{I_{h}^{2} (M_{h} - 1)}{d_{n}^{2} M_{h}} σ_{[h, h]} + \sum_{h}^{H} \sum_{h^{'} \neq h}^{H} \frac{I_{h} I_{h^{'}}}{d_{n}^{2}} σ_{[h, h^{'}]} . \end{array}$

It is clear that $I_{h} / (d_{n} M_{h})$ and $I_{h} I_{h^{'}} / d_{n}^{2},$ $h =1, \dots, H,$ are identically distributed. Using the equalities below

$\begin{array}{l} E (\frac{(M_{1} - 1) I_{1}^{2}}{d_{n}^{2} M_{1}}) & = E (\frac{I_{1}^{2}}{d_{n}^{2}}) - E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \\ E (\frac{I_{1} I_{2}}{d_{n}^{2}}) & = \frac{1}{H - 1} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}})) \end{array}$

we write

$\begin{array}{l} E {Var ({\hat{μ}}_{2} | R)} & = E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h]}^{2} + E (\frac{I_{1}^{2}}{d_{n}^{2}}) \sum_{h =1}^{H} σ_{[h, h]} - E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h, h]} \\ + \frac{1}{H - 1} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}})) \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} σ_{[h, h^{'}]} \\ = E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h]}^{2} + \frac{1}{H - 1} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}})) \sum_{h =1}^{H} \sum_{h =1}^{H} σ_{[h, h^{'}]} \\ + {E (\frac{I_{1}^{2}}{d_{n}^{2}}) - E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) - \frac{1}{H - 1} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}}))} \sum_{h =1}^{H} σ_{[h, h]} . \end{array} (A .4)$

We now show that $\sum_{h =1}^{H} \sum_{h^{'} =1}^{H} σ_{[h, h^{'}]} = H^{2} σ^{2} / (N - 1) .$ Using part (ii) of Lemma 1, we first observe that

$\begin{array}{l} \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} E (X_{[h]} X_{[h^{'}]}) & = \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} E (X_{(h)} X_{(h^{'})}) \\ =2 \sum_{i =1} \sum_{i < j} x_{i} x_{j} \sum_{h =1} \sum_{h^{'} =1} β (i, j, h, h^{'}) \\ =2 \sum_{i =1} \sum_{j > i} x_{i} x_{j} {\sum_{h =1}^{H} \sum_{λ =0}^{j - i - 1} \frac{(\begin{matrix} i - 1 \\ h - 1 \end{matrix}) (\begin{matrix} j - i - 1 \\ λ \end{matrix}) (\begin{matrix} N - j \\ H - λ - h \end{matrix})}{(\begin{matrix} N \\ H \end{matrix})} \\ \times \sum_{h^{'} =1}^{H} \frac{(\begin{matrix} j - 1 - h - λ \\ h^{'} - 1 \end{matrix}) (\begin{matrix} N - j - H + λ + h \\ H - h^{'} \end{matrix})}{(\begin{matrix} N - H \\ H \end{matrix})}} . \end{array}$

In the last sum of the above equation, we let $y = h^{'} - 1.$ After some simplification, we write

$\begin{array}{l} \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} E (X_{[h]} X_{[h^{'}]}) & =2 \sum_{i =1} \sum_{i < j} x_{i} x_{j} \sum_{h =1}^{H} \sum_{λ =0}^{j - i - 1} \frac{(\begin{matrix} i - 1 \\ h - 1 \end{matrix}) (\begin{matrix} j - i - 1 \\ λ \end{matrix}) (\begin{matrix} N - j \\ H - λ - h \end{matrix})}{(\begin{matrix} N \\ H \end{matrix})} \frac{H}{N - H} \\ = \frac{H}{N - H} \sum_{i =1} \sum_{j \neq i} x_{i} x_{j} \sum_{h =1}^{H} \frac{(\begin{matrix} i - 1 \\ h - 1 \end{matrix}) (\begin{matrix} N - i - 1 \\ H - h \end{matrix})}{(\begin{matrix} N \\ H \end{matrix})} . \end{array}$

Let $z = h - 1.$ The above expression reduces to

$\begin{array}{l} \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} E (X_{[h]} X_{[h^{'}]}) & = \frac{H}{N - H} \sum_{i =1} \sum_{j \neq i} x_{i} x_{j} \sum_{z =0}^{H - 1} \frac{(\begin{matrix} i - 1 \\ z \end{matrix}) (\begin{matrix} N - i - 1 \\ H - 1 - z \end{matrix})}{\frac{N (N - 1)}{H (H - N)} (\begin{matrix} N - 1 \\ H - 1 \end{matrix})} \\ = \frac{H^{2}}{N (N - 1)} \sum_{i =1} \sum_{j \neq i} x_{i} x_{j} = \frac{H^{2}}{N (N - 1)} {{(\sum_{i =1}^{N} x_{i})}^{2} - \sum_{i}^{N} x_{i}^{2}} . \end{array}$

Using the above equation, we conclude that

$\begin{array}{l} \sum_{h =1} \sum_{h^{'} =1} σ_{[h, h^{'}]} & = \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} E (X_{[h]} X_{[h^{'}]}) - \sum_{h =1} \sum_{h^{'} =1} E (X_{[h]} E X_{[h^{'}]}) \\ = \frac{H^{2}}{N (N - 1)} {{(\sum_{i =1}^{N} x_{i})}^{2} - \sum_{i}^{N} x_{i}^{2}} - \frac{H^{2}}{N^{2}} {(\sum_{i =1}^{N} x_{i})}^{2} \\ = - \frac{H^{2}}{N - 1} \frac{1}{N} \sum_{i =1}^{N} {(x_{i} - \bar{x})}^{2} = \frac{- H^{2}}{N - 1} σ^{2} . \end{array} (A .5)$

By inserting the above expression in equation (A.4), we write

$\begin{array}{l} E {Var ({\hat{μ}}_{2}) | R} & = E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h]}^{2} - \frac{H^{2} σ^{2}}{(H - 1) (N - 1)} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}})) \\ + {\frac{H}{H - 1} E (\frac{I_{1}^{2}}{d_{n}^{2}}) - E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) - \frac{1}{H (H - 1)}} \sum_{h =1}^{H} σ_{[h, h]} . \end{array} (A .6)$

We complete the proof by combining equations (A.3) and (A.6)

$\begin{array}{l} Var ({\hat{μ}}_{2}) & = \frac{H}{H - 1} Var (I_{1} / d_{n}) \sum_{h =1}^{H} {(μ_{[h]} - μ)}^{2} + E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) \sum_{h =1}^{H} σ_{[h]}^{2} \\ - \frac{H^{2} σ^{2}}{(H - 1) (N - 1)} (\frac{1}{H} - E (\frac{I_{1}^{2}}{d_{n}^{2}})) \\ + {\frac{H}{H - 1} E (\frac{I_{1}^{2}}{d_{n}^{2}}) - E (\frac{I_{1}^{2}}{d_{n}^{2} M_{1}}) - \frac{1}{(H - 1) H}} \sum_{h =1}^{H} σ_{[h, h]} . \end{array}$

Proof of Corollary 1: Proofs of $(i)$ and $(i i)$ are trivial. For the proof of (iii), we rewrite $\lim_{n \to \infty} n σ_{{\hat{μ}}_{2}}^{2}$ as

$\lim_{n \to \infty} n σ_{{\hat{μ}}_{2}}^{2} = \frac{1}{H} (\sum_{h =1}^{H} σ_{[h]}^{2} - \sum_{h =1}^{H} σ_{[h, h]}) - f σ^{2} .$

Using the equality (A.5), we write

$\begin{array}{l} - \frac{1}{H} \sum_{h =1}^{H} σ_{h, h} - f σ^{2} & \approx - \frac{N H}{N H^{2}} \sum_{h =1}^{H} σ_{[h, h]} + \frac{n (N - 1)}{N H^{2}} \sum_{h =1}^{H} \sum_{h^{'} =1}^{H} σ_{[h, h^{'}]} \\ = \sum_{h =1}^{H} σ_{[h, h]} \frac{N (n - H) - n}{N H^{2}} + \frac{n (N - 1)}{N H^{2}} \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} σ_{[h, h^{'}]} \\ \leq \frac{n (N - 1)}{N H^{2}} \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} σ_{[h, h^{'}]} \leq 0, \end{array}$

where $\approx$ is used to indicate approximate equality since we replace limit $f$ with its finite value for some $n .$ This inequality yields that

$\lim_{n \to \infty} n σ_{{\hat{μ}}_{2}}^{2} \leq \frac{1}{H} \sum_{h =1}^{H} σ_{[h]}^{2} = var (\sqrt{n} {\hat{μ}}_{0})$

which completes the proof.

Proof of Theorem 2: We first look at the expected value of $T_{1}$ and $T_{2}$ under design-0 and design-2. Using conditional expectation, the expected value of $T_{1}$ reduces to

$\begin{array}{l} E (T_{1}) & = \frac{1}{E (\frac{I_{1} I_{2}}{d_{n}^{2}})} \sum_{h =1}^{H} \sum_{h \neq h^{'}}^{H} E (\frac{I_{h} I_{h^{'}} M_{h} M_{h^{'}}}{M_{h} M_{h^{'}} d_{n}^{2}}) E {(X_{[h]} - X_{[h^{'}]})}^{2} \\ = {\begin{array}{l} 2 (H - 1) \sum_{h =1}^{H} σ_{[h]}^{2} + 2 H \sum_{h =1}^{H} {(μ_{[h]} - μ)}^{2} & for design- 0 \\ 2 (H - 1) \sum_{h =1}^{H} σ_{[h]}^{2} + 2 H \sum_{h =1}^{H} {(μ_{[h]} - μ)}^{2} - 2 \sum_{h =1}^{H} \sum_{h^{'} \neq h}^{H} σ_{[h, h^{'}]} & for design- 2. \end{array} \end{array}$

In a similar fashion, one can show that

$\begin{array}{l} E (T_{2}) & = \sum_{h =1}^{H} E (\frac{H I_{h}^{*} M_{h} (M_{h} - 1)}{M_{h} (M h - 1) d_{n}^{*}}) 2 E {(X_{[h] 1} - μ_{[h]}) - (μ_{[h]} - X_{[h] 2})}^{2} \\ = {\begin{array}{l} 2 H E (\frac{I_{1}^{*}}{d_{n}^{*}}) \sum_{h =1}^{H} σ_{[h]}^{2} & for design- 0 \\ 2 H E (\frac{I_{1}^{*}}{d_{n}^{*}}) {σ_{[h]}^{2} - \sum_{h}^{H} σ_{[h, h]}} & for design- 2. \end{array} \end{array}$

Since $I_{h}^{*} / d_{n}^{*},$ $h =1, \dots, H,$ are identically distributed, we have $E (I_{1}^{*} / d_{n}^{*}) = 1 / H .$ It follows that $E (T_{2}) =2 \sum_{h =1}^{H} σ_{[h]}^{2}$ for design-0 and $E (T_{2}) =2 \sum_{h =1}^{H} σ_{[h]}^{2} - 2 \sum_{h}^{H} σ_{[h, h]}$ for design-2.

For the proof of ${\bar{σ}}^{2}$ we first observe that $E (T_{1}) + E (T_{2}) =2 H^{2} σ^{2}$ for design-0 and $E (T_{1}) + E (T_{2}) = 2 N H^{2} σ^{2} / (N - 1)$ for design-2. The proofs then follows from these equalities.

We complete the proof of the unbiasedness of ${\hat{σ}}_{{\hat{μ}}_{0}}^{2}$ and ${\hat{σ}}_{{\hat{μ}}_{2}}^{2}$ and ${\bar{σ}}_{{\hat{μ}}_{2}}^{2}$ by inserting $E (T_{1})$ and $E (T_{2})$ in equations (2.5), (2.6) and (2.7).

The proofs for RSS estimators are similar. Hence, they are omitted.

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Appendix

References