Estimation and inference of domain means subject to qualitative constraints
Section 2. Constrained estimation and inference for domain means

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2.1  Notation and preliminaries

Let $U_N$ be the set of elements in a population of size $N.$ Consider a sample $s_N$ of size $n_N$ that is drawn from $U_N$ using a probability sampling design $p_N(\cdot ).$ Denote ${\pi }_{k\mathrm{,}N}\mathrm{<mo>=</mo>}\text{Pr}\left(k\in s_N\right)$ and ${\pi }_{kl\mathrm{,}N}\mathrm{<mo>=</mo>}\text{Pr}\left(k\in s_N\mathrm{,}l\in s_N\right)$ as the first and second order inclusion probabilities, respectively. Assume that ${\pi }_{k\mathrm{,}N}\mathrm{<mo>></mo>}\mathrm{0,}{\pi }_{kl\mathrm{,}N}\mathrm{<mo>></mo>}0$ for $k\mathrm{,}l\in U_N.$ To simplify notation, we will adopt the usual convention of suppressing the subscript $N$ unless it is needed for clarity. Denote ${\left\{U_d\right\}}_{d\mathrm{<mo>=</mo>}1}^D$ as a domain partition of $U,$ where $D$ is the number of domains and each $U_d$ is of size $N_d.$ Also, let $s_d$ be the subset of size $n_d$ of $s$ that belongs to $U_d.$

For any study variable $y,$ ${\overline{y}}_U\mathrm{<mo>=</mo>}{\left({\overline{y}}_{U_1},\dots ,{\overline{y}}_{U_D}\right)}^T$ denotes the vector of population domain means, where

$${\overline{y}}_{U_D}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in U_d}{y}_k}}{N_d}\mathrm{.}(2.1)$$

We will focus on the Hájek estimator of ${\overline{y}}_{U_D},$ given by

$${\tilde{y}}_{s_d}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in s_d}{y}_k/{\pi }_k}}{{\widehat{N}}_d}(2.2)$$

with ${\widehat{N}}_d\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s_d}}1/{\pi }_k,$ and let ${\tilde{y}}_s$ to be the vector of estimators. The results will also hold for the Horvitz-Thompson estimator with minor modifications, but it will not be explicitly addressed in what follows.

2.2  Proposed estimator

Assume there is information available regarding relationships between the population domain means that can be expressed with $m$ constraints through a $m\times D$  irreducible constraint matrix $A.$ A matrix $A$ is irreducible if none of its rows is a positive linear combination of other rows, and if the origin is also not a positive linear combination of its rows (Meyer, 1999). In practical terms, this means that there are no redundant constraints in $A.$ To take advantage of ${\tilde{y}}_s$ to obtain an estimator that respects these shape constraints, we propose the constrained estimator ${\tilde{\theta }}_s\mathrm{<mo>=</mo>}{\left({\tilde{\theta }}_{s_1}\mathrm{,}\dots \mathrm{,}{\tilde{\theta }}_{s_D}\right)}^T$ to be the unique vector that solves the following constrained weighted least squares problem,

$$\underset{\theta }{\min }{\left({\tilde{y}}_s-\theta \right)}^T{W}_s\left({\tilde{y}}_s-\theta \right)\text{subject}\text{to}A\theta \ge 0\mathrm{<mo>;</mo>}(2.3)$$

where $W_s$ is the diagonal matrix with elements ${\widehat{N}}_1/\widehat{N}\mathrm{,}{\widehat{N}}_2/\widehat{N}\mathrm{,}\dots \mathrm{,}{\widehat{N}}_D/\widehat{N},$ and $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{d\mathrm{<mo>=</mo>\; 1}}^D}{\widehat{N}}_d.$ The constrained problem in equation (2.3) can be alternatively written as finding the unique vector ${\tilde{\varphi }}_s$ that solves

$$\underset{\varphi }{\min }{\Vert {\tilde{z}}_s-\varphi \Vert }^2\text{subject}\text{to}{A}_s\varphi \ge 0\mathrm{,}(2.4)$$

where ${\tilde{z}}_s\mathrm{<mo>=</mo>}{W}_s^{1/2}{\tilde{y}}_s,$ $\varphi \mathrm{<mo>=</mo>}{W}_s^{1/2}\theta ,$ and $A_s\mathrm{<mo>=</mo>}A{W}_s^{-1/2}.$ The transformed constrained matrix $A_s$ is also irreducible if $A$ is, and it depends on the sample although $A$ does not. The solution ${\tilde{\varphi }}_s$ is the projection of ${\tilde{z}}_s$ onto the set of vectors $\varphi$ that satisfy the condition $A_s\varphi \ge 0.$ This set is a polyhedral convex cone, called the constraint cone ${\Omega }_s$ defined by $A_s;$ specifically,

$${\Omega }_s\mathrm{<mo>=</mo>}\left\{\varphi \in R^D\text{}\mathrm{<mo>:</mo>}{A}_s\varphi \ge 0\right\}\mathrm{.}(2.5)$$

We use the notation ${\tilde{\varphi }}_s\mathrm{<mo>=</mo>}\Pi \left({\tilde{z}}_s|{\Omega }_s\right),$ where $\Pi \left(u|S\right)$ stands for the projection of $u$ onto the set $S,$ i.e., the closest vector in $S$ to $u.$

Projections onto such cones are well understood; see Rockafellar (1970) or Meyer (1999) for details. In terms of this work, the main results from cone projection theory are summarized here. The cone can be characterized by a set of edges generating the cone; that is, a vector is in the cone if and only if it is a linear combination of the edges with non-negative coefficients. (Picture a pyramid with vertex at the origin, extending out indefinitely.) Subsets of the edges define the faces of the cone, and the projection of ${\tilde{z}}_s$ onto the cone lands on one of the faces. Once the edges defining this face are determined, the projection can be characterized as an ordinary least-squares projection onto the linear space spanned by this subset of edges. This property is crucial for both the algorithm for projection and for inference, because the projection onto the cone can be characterized as a linear projection.

For this work, we will project ${\tilde{z}}_s$ onto the polar cone ${\Omega }_s^0$ (Rockafellar, 1970, page 121), defined as

$${\Omega }_s^0\mathrm{<mo>=</mo>}\left\{\rho \in R^D\text{}\mathrm{<mo>:</mo>}\langle \rho \mathrm{,}\varphi \rangle \le \mathrm{0,}\forall \varphi \in {\Omega }_s\right\}\mathrm{,}(2.6)$$

where $\langle u\mathrm{,}v\rangle \mathrm{<mo>=</mo>}{u}^{T}v.$ That is, the polar cone is the set of vectors that form obtuse angles with all vectors in ${\Omega }_s.$ The polar cone is analogous to the orthogonal space in linear least-squares projections, in that the projection of a vector onto the polar cone is the residual of its projection onto the constraint cone, and vice-versa. Meyer (1999) showed that the negative rows of an irreducible matrix are the edges (generators) of the polar cone, leading to the following characterization of the polar cone in (2.6):

$${\Omega }_s^0\mathrm{<mo>=</mo>}\left\{\rho \in R^D\text{}\mathrm{<mo>:</mo>}\rho \mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{a}_j{\gamma }_{s_j}\text{}\mathrm{,}{a}_j\ge \mathrm{0,}j\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\}\mathrm{,}(2.7)$$

where ${\gamma }_{s_1}\text{}\mathrm{,}{\gamma }_{s_2}\text{}\mathrm{,}\dots \mathrm{,}{\gamma }_{s_m}$ are the rows of $-A_s.$ Robertson, Wright and Dykstra (1988, page 17) established necessary and sufficient conditions for a vector ${\tilde{\varphi }}_s$ to be the projection of ${\tilde{z}}_s$ onto ${\Omega }_s.$ That is, ${\tilde{\varphi }}_s\in {\Omega }_s$ solves the constrained problem in (2.4) if and only if

$$\langle {\tilde{z}}_s-{\tilde{\varphi }}_s\mathrm{,}{\tilde{\varphi }}_s\rangle \mathrm{<mo>=</mo>}\mathrm{0,}\text{and}\langle {\tilde{z}}_s-{\tilde{\varphi }}_s\mathrm{,}\varphi \rangle \le \mathrm{0,}\forall \varphi \in {\Omega }_s\mathrm{.}$$

Moreover, the above conditions can be adapted to the polar cone as follows: the vector ${\tilde{\rho }}_s\in {\Omega }_s^0$ minimizes ${\Vert {\tilde{z}}_s-\rho \Vert }^2$ over ${\Omega }_s^0$ if and only if

$$\langle {\tilde{z}}_s-{\tilde{\rho }}_s\mathrm{,}{\tilde{\rho }}_s\rangle \mathrm{<mo>=</mo>}\mathrm{0,}\text{and}\langle {\tilde{z}}_s-{\tilde{\rho }}_s\mathrm{,}{\gamma }_{s_j}\rangle \le 0\text{for}j\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\mathrm{.}(2.8)$$

The conditions in (2.8) can be used to show that the projection of ${\tilde{z}}_s$ onto the polar cone ${\Omega }_s^0$ coincides with the projection onto the linear space generated by the edges ${\gamma }_{s_j}$ such that $\langle {\tilde{z}}_s-{\tilde{\rho }}_s\mathrm{,}{\gamma }_{s_j}\rangle \mathrm{<mo>=</mo>}0.$ This set of edges could be empty, meaning that the projection onto ${\Omega }_s^0$ is equal to the projection onto the zero vector. In that case, the unconstrained minimum satisfies all the constraints. Alternatively, this set of edges might not be unique. To formalize these ideas, denote $V_{s\mathrm{,}J}\mathrm{<mo>=</mo>}\left\{{\gamma }_{s_j}\text{}\mathrm{<mo>:</mo>}j\in J\right\}$ for any $J\subseteq \left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\}.$ Define the set ${\overline{F}}_{s\mathrm{,}J}$ as,

$${\overline{F}}_{s\mathrm{,}J}\mathrm{<mo>=</mo>}\left\{\rho \in R^D\text{}\mathrm{<mo>:</mo>}\rho \mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in J}}{a}_j{\gamma }_{s_j}\text{}\mathrm{,}{a}_j\ge \mathrm{0,}j\in J\right\}\mathrm{,}(2.9)$$

where ${\overline{F}}_{s\mathrm{,}\varnothing }\mathrm{<mo>=</mo>}0$ by convention. (Technically, this set is the closure of a face of the cone.) That is, ${\overline{F}}_{s\mathrm{,}J}$ is a closed polyhedral sub-cone of ${\Omega }_s^0$ that starts at the origin and is defined by the edges in $V_{s\mathrm{,}J}.$ Further, let $L\left(V_{s\mathrm{,}J}\right)$ be the linear space generated by the vectors in $V_{s\mathrm{,}J}.$ It is shown in Meyer (1999) that projecting onto ${\Omega }_s^0$ is equivalent to projecting onto $L\left(V_{s\mathrm{,}J}\right),$ for an appropriate set $J.$ If the rows of the constraint matrix $A$ are linearly independent, then the minimal set $J$ is unique; otherwise there may be more than one $J$ that defines the linear space. In the latter case, however, the projection is still unique (see Theorem 1 of the next section).

Wu et al. (2016) considered the solution to (2.3), in the special case of a monotone relationship between domains defined along a single categorical variable. In that case, the solution is equivalent to that of the Pooled Adjacent Violator Algorithm (PAVA), which has an explicit expression in terms of a pooling of neighboring domains. The theoretical results in Wu et al. (2016) were obtained using that explicit expression, and hence do not apply to the more general setting considered here. Nevertheless, as was the case with the simple 6-domain example in Section 1 and in many situations of practical interest, the specific matrix $A$ will often correspond to a multivariate partial ordering of the domain means. Under partial ordering, the solution to the constrained minimization in (2.3) is again equivalent to a pooling of neighboring domains in such a way that the partial order constraints are respected. See for instance Robertson et al. (1988, page 23) for an explicit expression of this pooled domain expression under partial ordering, including the definition of the pooling. However, unlike PAVA in the univariate case, this does not lead to a practical general computational algorithm. In the current paper, we will allow for arbitrary irreducible constraint matrix $A,$ which will include partial ordering and univariate monotonicity as special cases.

One possible general approach to computing ${\tilde{\varphi }}_s$ is based on the edges of the constraint cone ${\Omega }_s.$ However, the number of edges can be considerably larger than the number of constraints for large values of $D,$ especially for the case when there are more constraints than domains (see Meyer, 1999). Moreover, given the lack of a general closed form solution for the edges of ${\Omega }_s$ (when $m\mathrm{<mo>></mo>}D),$ the edges need to be computed numerically in that case. This task is computationally demanding, which makes this approach an inefficient way to compute ${\tilde{\varphi }}_s.$ A more efficient algorithm based on computing the projection onto the polar cone has been developed: the Cone Projection Algorithm (CPA) (Meyer, 2013). This alternative approach takes advantage of the easy-to-find edges ${\gamma }_{s_j}$ of the polar cone, the conditions in (2.8), and the fact that $\Pi \left({\tilde{z}}_s|{\Omega }_s\right)\mathrm{<mo>=</mo>}{\tilde{z}}_s-\Pi \left({\tilde{z}}_s|{\Omega }_s^0\right).$ The latter fact is a key component on the proofs of the main theoretical results shown in this paper. CPA has been implemented in the software R into the coneproj package. See Liao and Meyer (2014) for further details.

For the situations in which the constraints correspond to complete or partial ordering, the CPA solution once again corresponds to domain pooling. After this, the domain mean estimates can be explicitly computed as sample-based domain means for the CPA-determined pooled domains. This greatly facilitates incorporating this methodology into survey estimation practice, because the pooled domain definitions can be readily communicated as part of the instructions accompanying a survey dataset release, and the estimates can be calculated without requiring access to specialized software.

2.3  Variance estimation of ${\tilde{\theta }}_{s_d}$

Estimating appropriately the variance of ${\tilde{\theta }}_{s_d}$ is a complicated task, derived from the fact that the projection of ${\tilde{z}}_s$ onto ${\Omega }_s^0$ (or onto ${\Omega }_s)$ might not always land on the same linear space $L\left(V_{s\mathrm{,}J}\right)$ for different samples $s.$ To better understand that, we define $G_s$ as the set of all subsets $J\subseteq \left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\}$ such that $\Pi \left({\tilde{z}}_s|{\Omega }_s^0\right)\mathrm{<mo>=</mo>}\Pi \left({\tilde{z}}_s|L\left(V_{s\mathrm{,}J}\right)\right)\in {\overline{F}}_{s\mathrm{,}J},$ as defined in (2.9). As noted earlier, there could be different sets $J_1$ and $J_2$ such that the projection onto the polar cone ${\Omega }_s^0$ is equal to projecting onto either $L\left(V_{s\mathrm{,}{J}_1}\right)$ or $L\left(V_{s\mathrm{,}{J}_2}\right).$ However, independently of which set is chosen, the projection ${\tilde{\rho }}_s$ is unique.

To illustrate the above point, consider the following restrictions when there are only 3 domains: the first domain mean is expected to be at the most equal to the second domain mean, and the third domain mean is expected to be at least equal to the average of the first two domain means. Hence, the constraint matrix $A$ can be expressed as

$$A\mathrm{<mo>=</mo>}\left(\begin{array}{rrr}\hfill -1& \hfill 1& \hfill 0\\ \hfill -1& \hfill -1& \hfill 2\end{array}\right)\mathrm{.}$$

Suppose it is observed that ${\tilde{y}}_{s_1}\mathrm{<mo>=</mo>}{\tilde{y}}_{s_2}\mathrm{<mo><</mo>}{\tilde{y}}_{s_3}.$ The transformed vector ${\tilde{z}}_s$ has elements of the form

$${\tilde{z}}_{s_1}\mathrm{<mo>=</mo>}\sqrt{\frac{{\widehat{N}}_1}{\widehat{N}}}{\tilde{y}}_{s_1}\mathrm{,}{\tilde{z}}_{s_2}\mathrm{<mo>=</mo>}\sqrt{\frac{{\widehat{N}}_2}{\widehat{N}}}{\tilde{y}}_{s_2},{\tilde{z}}_{s_3}\mathrm{<mo>=</mo>}\sqrt{\frac{{\widehat{N}}_3}{\widehat{N}}}{\tilde{y}}_{s_3}\mathrm{.}$$

In this setting, it is straightforward to see that $\Pi \left({\tilde{z}}_s|{\Omega }_s^0\right)\mathrm{<mo>=</mo>}0.$ In the process of computing it using the general algorithm, we project ${\tilde{z}}_s$ onto each of the $2^2\mathrm{<mo>=</mo>}4$ linear spaces generated by the polar cone edges

$${\gamma }_{s_1}\mathrm{<mo>=</mo>}{\left(\sqrt{\frac{\widehat{N}}{{\widehat{N}}_1}}\mathrm{,}-\sqrt{\frac{\widehat{N}}{{\widehat{N}}_2}}\mathrm{,}0\right)}^T\mathrm{,}{\gamma }_{s_2}\mathrm{<mo>=</mo>}{\left(\sqrt{\frac{\widehat{N}}{{\widehat{N}}_1}}\mathrm{,}\sqrt{\frac{\widehat{N}}{{\widehat{N}}_2}}\mathrm{,}-2\sqrt{\frac{\widehat{N}}{{\widehat{N}}_3}}\right)}^T\mathrm{.}$$

Hence, it can be seen that the conditions $\Pi \left({\tilde{z}}_s|{\Omega }_s^0\right)\mathrm{<mo>=</mo>}0\mathrm{<mo>=</mo>}\Pi \left({\tilde{z}}_s|L\left(V_{s\mathrm{,}J}\right)\right)\in {\overline{F}}_{s\mathrm{,}J}$ are satisfied only for $J\mathrm{<mo>=</mo>}\varnothing$ and $J\mathrm{<mo>=</mo>}\left\{1\right\},$ which implies that $G_s\mathrm{<mo>=</mo>}\left\{\varnothing \mathrm{,}\left\{1\right\}\right\}.$ Moreover, note that $V_{s\mathrm{,}\varnothing }$ and $V_{s\mathrm{,}\left\{1\right\}}$ do not span the same linear spaces, which is what complicates the variance estimation of ${\tilde{\theta }}_{s_d}.$ In the model-based case with continuous variables, the set of sample vectors where these scenarios occur has measure zero. However, they cannot be excluded in the design-based setting.

We propose a variance estimator for ${\tilde{\theta }}_{s_d}$ that relies on the sets in $G_s$ and is based on linearization methods. Consider any fixed set $J\in G_s,$ and let $P_{s\mathrm{,}J}$ be the projection matrix corresponding to the linear space $L\left(V_{s\mathrm{,}J}\right),$ where $P_{s\mathrm{,}\varnothing }$ is the matrix of zeros by convention. By the selection of $J,$ then ${\tilde{\rho }}_s$ can be expressed as $P_{s\mathrm{,}J}{\tilde{z}}_s,$ which implies that ${\tilde{\theta }}_s$ can be written as ${\tilde{\theta }}_{s\mathrm{,}J}\mathrm{<mo>=</mo>}{\tilde{y}}_s-W_s^{-1/2}{P}_{s\mathrm{,}J}{W}_s^{1/2}{\tilde{y}}_s,$ where we add the subscript $J$ in ${\tilde{\theta }}_s$ to be aware that the expression depends on the chosen $J.$

Now, observe that ${\tilde{\theta }}_{s\mathrm{,}J}$ is a smooth non-linear function of the ${\widehat{t}}_d’s$ and the ${\widehat{N}}_d’s,$ where ${\widehat{t}}_d$ is the Horvitz-Thompson estimator of $t_d\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in U_d}}{y}_k.$ Therefore, treating $J$ as fixed, we obtain the asymptotic variance of ${\tilde{\theta }}_{s_d\mathrm{,}J}$ via Taylor linearization (Särndal et al., 1992, page 175) as

$$\text{AV}\left({\tilde{\theta }}_{s_d\mathrm{,}J}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}}{\displaystyle \sum _{l\in U}}{\Delta }_{kl}\frac{u_k}{{\pi }_k}\frac{u_l}{{\pi }_l}\mathrm{,}(2.10)$$

where ${\Delta }_{kl}\mathrm{<mo>=</mo>}{\pi }_{kl}-{\pi }_k{\pi }_l,$ and

$$u_k\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\alpha }_i{y}_k{1}_{k\in U_i}+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\beta }_i{1}_{k\in U_i}\text{for}k\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\mathrm{,}$$

with $1_A$ being the indicator variable for the event $A,$ and

$${\alpha }_i\mathrm{<mo>=</mo>}\frac{\partial {\tilde{\theta }}_{s_d\mathrm{,}J}}{\partial {\widehat{t}}_i}|{}_{\begin{array}{c}\left({\widehat{t}}_1,\dots ,{\widehat{t}}_D,{\widehat{N}}_1,\dots ,{\widehat{N}}_D\right)=\left(t_1,\dots ,t_D,N_1,\dots ,N_D\right)\end{array}}\mathrm{<mo>;</mo>}{\beta }_i\mathrm{<mo>=</mo>}\frac{\partial {\tilde{\theta }}_{s_d\mathrm{,}J}}{\partial {\widehat{N}}_i}|{}_{\left({\widehat{t}}_1,\dots ,{\widehat{t}}_D,{\widehat{N}}_1,\dots ,{\widehat{N}}_D\right)=\left(t_1,\dots ,t_D,N_1,\dots ,N_D\right)}.$$

In addition, a consistent estimator of the asymptotic variance in (2.10) is given by

$$\widehat{V}\left({\tilde{\theta }}_{s_d\mathrm{,}J}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}}{\displaystyle \sum _{l\in s}}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{{\widehat{u}}_k}{{\pi }_k}\frac{{\widehat{u}}_l}{{\pi }_l}\mathrm{,}(2.11)$$

where

$${\widehat{u}}_k\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>}1}^D}{\widehat{\alpha }}_i{y}_k{1}_{k\in s_i}+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\widehat{\beta }}_i{1}_{k\in s_i}\text{for}k\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\mathrm{,}$$

with ${\widehat{\alpha }}_i,{\widehat{\beta }}_i$ obtained from ${\alpha }_i\mathrm{,}{\beta }_i$ by substituting the appropriate Horvitz-Thompson estimators for each population total. We propose the estimator in (2.11), computed at the $J$ obtained in the sample, as a variance estimator of ${\tilde{\theta }}_{s_d}.$

To provide a clear example of the proposed variance estimator for ${\tilde{\theta }}_{s_d},$ consider the setting presented at the beginning of this subsection. Since $G_s\mathrm{<mo>=</mo>}\left\{\varnothing \mathrm{,}\left\{1\right\}\right\},$ it might be of interest to compute the estimated variance of ${\tilde{\theta }}_{s_d\mathrm{,}J}$ for $J\mathrm{<mo>=</mo>}\left\{1\right\}$ and certain $d.$ The matrix $P_{s\mathrm{,}\left\{1\right\}}$ is the projection matrix corresponding to the linear space generated by ${\gamma }_{s_1},$ given by

$$P_{s\mathrm{,}\left\{1\right\}}\mathrm{<mo>=</mo>}{\left({\widehat{N}}_1+{\widehat{N}}_2\right)}^{-1}\left(\begin{array}{ccc}{\widehat{N}}_2& -\sqrt{{\widehat{N}}_1{\widehat{N}}_2}& 0\\ -\sqrt{{\widehat{N}}_1{\widehat{N}}_2}& {\widehat{N}}_1& 0\\ 0& 0& 0\end{array}\right)\mathrm{.}$$

Note that $P_{s\mathrm{,}\left\{1\right\}}$ is a function of $\left({\widehat{N}}_1,{\widehat{N}}_2,{\widehat{N}}_3\right)$ because ${\gamma }_{s_1}$ is. Using the above equation, ${\tilde{\theta }}_{s\mathrm{,}\left\{1\right\}}$ can be simplified to the following expression,

$$\begin{array}{ll}{\tilde{\theta }}_{s\mathrm{,}\left\{1\right\}}\mathrm{<mo>=</mo>}{\left({\tilde{\theta }}_{s_1,\left\{1\right\}},{\tilde{\theta }}_{s_2,\left\{1\right\}},{\tilde{\theta }}_{s_3,\left\{1\right\}}\right)}^T\hfill & ={\left(\frac{{\widehat{N}}_1{\tilde{y}}_{s_1}+{\widehat{N}}_2{\tilde{y}}_{s_2}}{{\widehat{N}}_1+{\widehat{N}}_2}\mathrm{,}\frac{{\widehat{N}}_1{\tilde{y}}_{s_1}+{\widehat{N}}_2{\tilde{y}}_{s_2}}{{\widehat{N}}_1+{\widehat{N}}_2}\mathrm{,}{\tilde{y}}_{s_3}\right)}^T\hfill \\ \hfill & ={\left(\frac{{\widehat{t}}_1+{\widehat{t}}_2}{{\widehat{N}}_1+{\widehat{N}}_2}\mathrm{,}\frac{{\widehat{t}}_1+{\widehat{t}}_2}{{\widehat{N}}_1+{\widehat{N}}_2}\mathrm{,}\frac{{\widehat{t}}_3}{{\widehat{N}}_3}\right)}^T\mathrm{.}\hfill \end{array}$$

Therefore, given a domain $d,$ the $\alpha ’s$ and $\beta ’s$ can be derived by taking the partial derivatives of ${\tilde{\theta }}_{s_d\mathrm{,}\left\{1\right\}}$ with respect to the $\widehat{t}’s$ and $\widehat{N}’s,$ and evaluating such derivatives at the $t’s$ and $N’s.$ For $d\mathrm{<mo>=</mo>}2,$ that is,

$$\begin{array}{llll}{\alpha }_1\hfill & \mathrm{<mo>=</mo>}{\alpha }_2\hfill & =\frac{1}{N_1+N_2} \mathrm{,}\hfill & {\alpha }_3\mathrm{<mo>=</mo>}\mathrm{0,}\hfill \\ {\beta }_1\hfill & \mathrm{<mo>=</mo>}{\beta }_2\hfill & =-\frac{t_1+t_2}{{\left(N_1+N_2\right)}^2}\mathrm{,}\hfill & {\beta }_3\mathrm{<mo>=</mo>}0.\hfill \end{array}$$

The $\widehat{\alpha }’s$ and $\widehat{\beta }’s$ are computed by substituting Horvitz-Thompson estimators in the above equations, which are then used to evaluate ${\widehat{u}}_k$ for each $k$ in the sample $s.$ Finally, the proposed variance estimator in (2.11) can be computed.

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