Statistical inference based on judgment post-stratified samples in finite population Section 3. Rao-Blackwellized estimatorStatistical inference based on judgment post-stratified samples in finite population Section 3. Rao-Blackwellized estimator

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In this section, we construct estimators that improve the performance of the JPS estimators in the previous section. For a given simple random sample $X_{s_1}\mathrm{,}\dots \mathrm{,}{X}_{s_n},$ the sets $S_{j\mathrm{,}H};$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n\mathrm{,}$ can be constructed over all possible matching of $n\left(H-1\right)$ additional units to $n$ fully measured units. Each construction creates a new set of ranks hence a new estimate. We combine all these estimates by using Rao-Blackwell theorem. Let

$$\begin{array}{ll}{\tilde{\mu }}_r\hfill & \mathrm{<mo>=</mo>}E\left(\widehat{\mu }|X\right)\mathrm{<mo>=</mo>}E\left\{{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{I_h}{M_h{d}_n}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^n}{X}_{j}I\left(R_j\mathrm{<mo>=</mo>}h\right)|X_1\mathrm{,}\dots \mathrm{,}{X}_n\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{X}_{i}E\left[{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{I_{h}I\left(R_i\mathrm{<mo>=</mo>}h\right)}{M_h{d}_n}|X\right]\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{X}_i{a}_{i|X}\mathrm{<mo>;</mo>}r\mathrm{<mo>=</mo>\; 0,2,}\hfill \end{array}$$

where

$$a_{i|X}\mathrm{<mo>=</mo>}E\left[{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{I_{h}I\left(R_i\mathrm{<mo>=</mo>}h\right)}{M_h{d}_n}|X\right]\mathrm{.}$$

The expectation in $a_{i|X}$ is taken over the conditional distributions of $R_j;$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ $M_h;$ $h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}N,$ and $d_n$ given $X.$ We note that ranks, $R_j;$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ are assigned independently in each set $S_{j\mathrm{,}H}.$ Hence the joint distributions of $R_j;$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ given the measured observations $X_j;$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ are all independent

$${\alpha }_{h_1,\dots \mathrm{,}{h}_n|X}\mathrm{<mo>=</mo>}P\left(R_1\mathrm{<mo>=</mo>}{h}_1\mathrm{,}\dots \mathrm{,}{R}_n\mathrm{<mo>=</mo>}{h}_n|X\right)\mathrm{<mo>=</mo>}{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^n}P\left(R_j\mathrm{<mo>=</mo>}{h}_j|X\right)\mathrm{,}1\le h_j\le H\mathrm{.}$$

Assume that population ranks, $s_j;$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n,$ of sample units are available, To construct the conditional distribution of $R_j\mathrm{<mo>=</mo>}{h}_j$ given $X_j\mathrm{<mo>=</mo>}{x}_{s_j},$ we first observe that

$$P\left(R_j\mathrm{<mo>=</mo>}{h}_j\mathrm{,}{X}_j\mathrm{<mo>=</mo>}{x}_{s_j}\right)\mathrm{<mo>=</mo>}\beta \left(s_j\mathrm{,}{h}_j\right)/H\text{and}P\left(X_j\mathrm{<mo>=</mo>}{x}_{s_j}\right)\mathrm{<mo>=</mo>}1/N\mathrm{.}$$

Using these joint and marginal probability mass functions, we write

$${\alpha }_{h_j|s_j}\mathrm{<mo>=</mo>}P\left(R_j\mathrm{<mo>=</mo>}{h}_j|X\right)\mathrm{<mo>=</mo>}P\left(R_j\mathrm{<mo>=</mo>}{h}_j|X_j\mathrm{<mo>=</mo>}{x}_{s_j}\right)\mathrm{<mo>=</mo>}\frac{\left(\begin{array}{c}{s}_j-1\\ h_j-1\end{array}\right)\left(\begin{array}{c}N-s_j\\ H-h_j\end{array}\right)}{\left(\begin{array}{c}N-1\\ H-1\end{array}\right)},h_j=1,\dots ,H,x_{s_j}\in \mathcal{P},(3.1)$$

where $x_{s_j}$ is the $s_j^{\text{th}}$ smallest unit in the population. The evaluation of $a_{s_j|X}$ in ${\tilde{\mu }}_r$ is computationally intensive. Even though the conditional distributions of rank $R_j’\text{s}$ are independent for given $X,$ they are not identically distributed. Hence, the conditional distribution of judgment class sample size vector $M$ given $X$ does not have a multinomial distribution.

We now introduce an approximation to evaluate ${\tilde{\mu }}_r.$ We first recognize that the conditional distribution of $R_j\mathrm{<mo>=</mo>}h$ given $X_j\mathrm{<mo>=</mo>}{x}_{s_j}$ is a hypergeometric distribution. Thus we can generate $R_j$ from this hypergeometric distributions for given values of $X.$

Algorithm 1.

• I.    Select an integer $B.$  For $b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B,$  generate $R^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}\left\{R_1^{b\mathrm{,\; <mo>*</mo>}}\mathrm{,}\dots \mathrm{,}{R}_n^{b\mathrm{,\; <mo>*</mo>}}\right\}$  from ${\alpha }_{R_j|X_j\mathrm{<mo>=</mo>}{x}_{s_j}}$  in equation (3.1).
• II.   Using $R^{b\mathrm{,\; <mo>*</mo>}}$  compute $I_h^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}I\left(M^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>></mo>\; 0}\right),$ $M_h^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^n}I\left(R_j^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}h\right),$ $d_n^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{h\mathrm{<mo>=</mo>\; 1}}^H}{I}_h^{b\mathrm{,\; <mo>*</mo>}}$  and

$$a_i^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{I^{b\mathrm{,\; <mo>*</mo>}}}{M_h^{b\mathrm{,\; <mo>*</mo>}}{d}_n^{b\mathrm{,\; <mo>*</mo>}}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}I\left(R_i^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}h\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$$

We approximate $a_i$ with ${\overline{a}}_i^{\mathrm{*}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B}{a}_i^{b\mathrm{,\; <mo>*</mo>}}/B,$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$ From law of large numbers ${\overline{a}}_i^{\mathrm{*}}$ approaches to $a_i$ as $B$ gets large. Rao-Blackwellized estimator ${\tilde{\mu }}_r$ is then approximated by ${\tilde{\mu }}_r^{\mathrm{*}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{\overline{a}}_i^{\mathrm{*}}{X}_i.$

If the population ranks of sample units are not available, Algorithm 1 may not be usable. In this case, we use the collection of all unmeasured units to construct Rao-Blackwellized estimators. Let $Y^{{\rm T}}\mathrm{<mo>=</mo>}\left(Y_1\mathrm{,}\dots \mathrm{,}{Y}_{n\left(H-1\right)}\right)$ be the auxiliary variables on unmeasured random variables. We use the following algorithm to approximate the Rao-Blackwellized estimators.

Algorithm 2. For $b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B,$  repeat the steps I-IV.

• I.    Perform a random permutation on the entries of vector $Y$  to obtain $Y_b\mathrm{<mo>=</mo>}permute\left(Y\right).$
• II.   Divide the entries of $Y_b$  into $n$  sets, each of size $H-1.$
• III. Match these $n$  sets of size $H-1$  with $n$ $\mathrm{Y-}$values of the measured units to form $n$  sets, each of size $H.$  Obtain the rank of the $X$  measurement from the rank of corresponding $Y$  value in each set, $R_b^{\mathrm{*}}\mathrm{<mo>=</mo>}\left(R_{b\mathrm{,1}}^{\mathrm{*}}\mathrm{,}\dots \mathrm{,}{R}_{b\mathrm{,}n}^{\mathrm{*}}\right).$
• IV.  Using $R_b^{\mathrm{*}}$  compute $I_h^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}I\left(M^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>></mo>\; 0}\right),$ $M_h^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^n}I\left(R_{b\mathrm{,}j}^{\mathrm{*}}\mathrm{<mo>=</mo>}h\right),$ $d_n^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{h\mathrm{<mo>=</mo>\; 1}}^H}{I}_h^{b\mathrm{,\; <mo>*</mo>}}$ and

$$a_i^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{I^{b\mathrm{,\; <mo>*</mo>}}}{M_h^{b\mathrm{,\; <mo>*</mo>}}{d}_n^{b\mathrm{,\; <mo>*</mo>}}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}I\left(R_i^{b\mathrm{,\; <mo>*</mo>}}\mathrm{<mo>=</mo>}h\right)\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$$

• V.   Compute the Rao-Blackwellized estimator ${\tilde{\mu }}_r\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^n}{\overline{a}}_i^{\mathrm{*}}{X}_i,$ $r\mathrm{<mo>=</mo>\; 0,}2,$  where ${\overline{a}}_i^{\mathrm{*}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B}{a}_i^{b\mathrm{,\; <mo>*</mo>}}/B.$

Even though large values of $B$ provides better approximation to Rao-Blackwellized estimators, it may require additional computational effort and may not be feasible in practice. On the other hand, even small values of $B,$ such as $B\mathrm{<mo>=</mo>\; 5}$ could provide a significant improvement.

We now consider constructing estimators for the variance of Rao-Blackwellized estimators. Obtaining analytic expressions for the variances of ${\tilde{\mu }}_r$ is a challenge. Difficulty arises from the fact that there is no analytic expressions for the computation of $a_{i|X};$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n.$ We then appeal to a bootstrap procedure to compute the variance of ${\tilde{\mu }}_r.$ Bootstrap estimators can be constructed from a plug-in method. Let $\theta$ be a statistical functional $\theta \mathrm{<mo>=</mo>}T\left(\mathcal{P}\right),$ where $\mathcal{P}$ is the finite population. The bootstrap estimate of $\theta$ can be obtained from $\widehat{\theta }\mathrm{<mo>=</mo>}T\left(\widehat{\mathcal{P}}\right),$ where $\widehat{\mathcal{P}}$ is the empirical population. In finite population setting, the construction of the empirical population plays an important role to preserve the without replacement policies of bootstrap samples. Let $K$ be the integer part of $N/n$ and $k\mathrm{<mo>=</mo>}N-Kn.$ We construct $\widehat{\mathcal{P}}$ with

$$\widehat{\mathcal{P}}\mathrm{<mo>=</mo>}\left\{\underset{K\text{times}}{\underbrace{X\mathrm{,}\dots \mathrm{,}X}}\mathrm{,}{X}_{t_1}\mathrm{,}\dots \mathrm{,}{X}_{t_k}\right\}\mathrm{,}$$

where $X_{t_j};$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k\mathrm{,}$ are selected at random from $X\mathrm{<mo>=</mo>}\left\{X_{s_1}\mathrm{,}\dots \mathrm{,}{X}_{s_n}\right\}.$ With this construction, the population size, $N,$ is the same in both $\widehat{\mathcal{P}}$ and $\mathcal{P}.$ We generate bootstrap re-samples $X^{\mathrm{<mo>*</mo>\; ,1}}\mathrm{<mo>=</mo>}\left\{X_{s_1}^{\mathrm{<mo>*</mo>\; ,1}}\mathrm{,}\dots \mathrm{,}{X}_{s_n}^{\mathrm{<mo>*</mo>\; ,1}}\right\}$ from $\widehat{\mathcal{P}}$ with replacement for design-0 and without replacement for design-2. To construct the bootstrap distribution of the estimator ${\tilde{\mu }}_r,$ we generate re-samples $X^{\mathrm{<mo>*</mo>\; ,}c}\mathrm{,}c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C$ and compute

$${\tilde{\mu }}_r^{\mathrm{<mo>*</mo>\; ,}c}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^n}{\overline{a}}_i^{\mathrm{*}}{X}_{s_i}^{\mathrm{<mo>*</mo>\; ,}c}\mathrm{,}c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C$$

from Algorithm 1 or 2. The bootstrap variance estimate of ${\tilde{\mu }}_r$ is then obtained from

$${\widehat{\sigma }}_{{\tilde{\mu }}_r}^2\mathrm{<mo>=</mo>}\frac{1}{C-1}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^C}{\left\{{\tilde{\mu }}_r^{\mathrm{<mo>*</mo>\; ,}c}-{\tilde{\mu }}_r^{\mathrm{*}}\right\}}^2\mathrm{,}(3.2)$$

where ${\tilde{\mu }}_r^{\mathrm{*}}$ is the mean of ${\tilde{\mu }}_r^{\mathrm{<mo>*</mo>\; ,}c}\mathrm{,}c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C.$

A bootstrap $\left(1-\gamma \right)\mathrm{100\; <mi>\%</mi>}$ percentile confidence interval for $\mu$ is constructed by $\left(L_r^{\gamma /2}\mathrm{,}{L}_r^{1-\gamma /2}\right),$ where $L_r^a$ is the $a^{\text{th}}$ quantiles of ${\tilde{\mu }}_r^{\mathrm{*}}$ satisfying $P\left({\tilde{\mu }}_r^{\mathrm{*}}\le L_r^a|\widehat{\mathcal{P}}\right)$ for $\mathrm{0\; <mo><</mo>}a\mathrm{<mo><</mo>\; 1.}$

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