Estimation and inference of domain means subject to qualitative constraints
Section 6. Conclusions

Table of contents

We have proposed a general methodology to estimate domain means which makes it possible incorporate natural restrictions between domains into design-based estimation. It was shown to improve estimation and inference, especially on small domains. As this new methodology covers a broad range of shape assumptions beyond univariate monotonicity, it aims to jointly take advantage of several types of qualitative information that arises naturally for survey data. Additional shapes that may be imposed include convexity or log-concavity; the latter might be imposed if the population domain means are believed to be increasing and then decreasing over a set of domains. Future work by the authors will include a “relaxed monotone” estimator to be used when the population domain means are “roughly” monotone in some sequence of domains. For the relaxed monotone estimator, a type of moving average over the domains is used to implement the constraints, allowing the estimator to have some departures from monotonicity.

We also proposed a design-based variance estimation method of the estimator, which only requires knowledge of the sample-specific constraint set. Replication-based methods are shown to behave similarly. From the computational side, the estimator is based on the Cone Projection Algorithm which is efficiently implemented in the package coneproj and freely available. In the important practical case of partial ordering, the constrained estimator is equivalent to a pooling of neighboring domains, so that once the constraint set is identified by CPA, subsequent computations of estimators and variance estimators can be done directly using traditional design-based estimation for the relevant domains.

An important practical issue, as illustrated in the NSCG analysis in Section 5, is the determination of when the imposed constraint might not be valid for a particular survey application. Recently, Oliva-Aviles, Meyer and Opsomer (2019) proposed the sample-based Cone Information Criterion as a criterion to choose between the constrained and unconstrained fits for the estimator of Wu et al. (2016). That approach is generalizable to the setting considered here, and is currently under development.

Appendix

The first part of this appendix contains lemmas used to obtain the theoretical results discussed in this paper. Proofs of the theorems are included at the end of this appendix.

Lemma 1. If a non-zero vector can be written as the positive linear combination of linearly dependent non-zero vectors, then it can be expressed as the positive linear combination of a linearly independent subset of these.

Proof. Let $v$ be a non-zero vector such that it can be written as $v = \sum_{i =1}^{k} a_{i} l_{i}$ where $l_{1}, l_{2}, \dots, l_{k}$ are non-zero vectors and $a_{i} > 0$ for $i = 1, 2, \dots, k .$ If this set of vectors is not linearly independent, then there exist constants $b_{1}, \dots, b_{k},$ not all zero, such that $\sum_{i =1}^{k} b_{i} l_{i} = 0,$ and for any $c \in R,$ $v = \sum_{i =1}^{k} (a_{i} + c b_{i}) l_{i} .$ Let $c = - \min_{i : b_{i} \neq 0} a_{i} / b_{i};$ then $a_{i} + c b_{i} \geq 0$ for $i = 1, \dots, k$ but for at least one $i,$ $a_{i} + c b_{i} = 0.$ Then we have written $v$ as a positive linear combination of a proper subset of the vectors. If this subset is still linearly dependent, the process can be repeated.

Lemma 2. If $A$ is a $m \times D$ irreducible matrix and $B$ is a $D \times D$ nonsingular matrix, then $\tilde{A} = A B$ is also irreducible.

Proof. Suppose ${\tilde{A}}^{T} c = 0$ for some $c \in R^{m},$ $c \geq 0 .$ Then $B^{T} A^{T} c = 0$ implies that $A^{T} c = 0$ by the non-singularity of $B .$ Because $A$ is irreducible, we must have $c = 0,$ so the origin is not a positive linear combination of rows of $\tilde{A} .$ Next, suppose that one of the rows of $\tilde{A}$ is a positive linear combination of other rows of $\tilde{A} .$ This means we can write ${\tilde{A}}^{T} b = 0,$ where $b_{j} = - 1$ for some $j \in {1, \dots, m}$ and $b_{i} \geq 0,$ $i \neq j .$ But ${\tilde{A}}^{T} b = 0$ implies that $B^{T} A^{T} b = 0$ implies that $A^{T} b = 0$ by the non-singularity of $B .$ We can’t have $A^{T} b = 0$ for this $b,$ so we can’t have a row of $\tilde{A}$ is a positive linear combination of other rows of $\tilde{A} .$ Therefore, $\tilde{A}$ is irreducible.

Lemma 3. Let $A$ be a $m \times D$ matrix. Also, let $S_{1}$ and $S_{2}$ be $D \times D$ diagonal matrices with nonzero elements on the diagonal. For any set $J \subseteq {1, 2, \dots, m},$ denote $V_{i, J}$ to be the set of vectors in rows $J$ of $A_{i} = A S_{i},$ $i = 1, 2.$ Then, for any $J^{*} \subseteq J,$

$L (V_{1, J^{*}}) = L (V_{1, J}) \Leftrightarrow L (V_{2, J^{*}}) = L (V_{2, J}) .$

Proof. Let $A_{i, J} = A_{J} S_{i},$ $i = 1, 2;$ where $A_{J}$ denotes the submatrix of $A$ that contains the rows in positions $J .$ First, assume that $L (V_{1, J^{*}}) = L (V_{1, J}) .$ Since $J^{*} \subseteq J,$ it is straightforward to see that $L (V_{2, J^{*}}) \subseteq L (V_{2, J}) .$ Now, consider any $v \in L (V_{2, J})$ so that $v = A_{2, J}^{T} a = S_{2} A_{J}^{T} a$ for some vector $a .$ Then, we have $S_{1} S_{2}^{- 1} v = S_{1} A_{J}^{T} a \in L (V_{1, J}) .$ By assumption, there exists a vector $b$ such that $S_{1} S_{2}^{- 1} v = S_{1} A_{J^{*}}^{T} b .$ Therefore, $v = S_{2} A_{J^{*}}^{T} b \in L (V_{2, J^{*}}) .$ Thus, $L (V_{2, J}) \subseteq L (V_{2, J^{*}}) .$ Analogously, it follows that $L (V_{2, J^{*}}) = L (V_{2, J})$ implies $L (V_{1, J^{*}}) = L (V_{1, J}) .$

Lemma 4. Under Assumptions A1-A5, the following statements hold:

(i)

The $N^{- 1} {\hat{t}}_{d}$ are uniformly bounded.

(ii)

The $N^{- 1} {\hat{N}}_{d}$ are uniformly bounded above and uniformly bounded away from zero.

(iii)

var $(N^{- 1} {\hat{t}}_{d}) = O (n^{- 1})$ and var $(N^{- 1} {\hat{N}}_{d}) = O (n^{- 1}) .$

(iv)

$E [{(N^{- 1} {\hat{t}}_{d} - r_{d} μ_{d})}^{2}] = O (n^{- 1})$ and $E [{(N^{- 1} {\hat{N}}_{d} - r_{d})}^{2}] = O (n^{- 1}) .$

Proof.

(i)

Note that

$\frac{| {\hat{t}}_{d} |}{N} = | \frac{\sum_{k \in s_{d}} y_{k} / π_{k}}{N} | \leq \frac{\sum_{k \in U} | y_{k} |}{λ N}$

which does not depend on

s,

and is bounded independently of

N

by Assumption A2.

(ii)

From Assumptions A4 and A5, note that

$\frac{ε n}{D N} \leq \frac{n_{d}}{N} \leq \frac{{\hat{N}}_{d}}{N} = N^{- 1} \sum_{k \in s_{d}} 1 / π_{k} \leq λ^{- 1} N^{- 1} N_{d} \leq λ^{- 1},$

where both lower and upper bounds do not depend on

s,

and are bounded for all

N

by Assumptions A1 and A4.

(iii)

Note that

$n var (N^{- 1} {\hat{t}}_{d}) = n var (N^{- 1} \sum_{k \in s_{d}} y_{k} / π_{k}) \leq \frac{\sum_{k \in U_{d}} y_{k}^{2}}{λ^{2} N} (\frac{n}{N} + n \max_{k, l \in U_{d} : k \neq l} | Δ_{k l} |)$

which is bounded by Assumptions A2, A4 and A5. Setting

y_{k} \equiv 1

and following an analogous argument, it can be shown that

n var (N^{- 1} {\hat{N}}_{d}) = O (1) .

(iv)

Since

$E [{(N^{- 1} {\hat{t}}_{d} - r_{d} μ_{d})}^{2}] = var (N^{- 1} {\hat{t}}_{d}) + {(\frac{N_{d}}{N} {\bar{y}}_{U_{d}} - r_{d} μ_{d})}^{2},$

Assumption A3 and (iii) lead to the desired conclusion. Analogously, we find

$E [{(N^{- 1} {\hat{N}}_{d} - r_{d})}^{2}] = O (n^{- 1}) .$

Proof of Theorem 1. First, suppose that $Π (z | Ω^{0}) = Π (z | L (V_{J})) = 0 .$ In that case, any subset $J^{*} \subset J$ such that $V_{J}$ is linearly independent will satisfy $Π (z | L (V_{J^{*}})) = 0 \in {\bar{F}}_{J^{*}} .$ Hence, it is enough to choose $J^{*} \subset J$ such that $V_{J^{*}}$ is linearly independent and spans $L (V_{J}) .$ Now, suppose that $Π (z | Ω^{0}) \neq 0 .$ Since $Π (z | Ω^{0}) = Π (z | L (V_{J})) \in {\bar{F}}_{J},$ $Π (z | L (V_{J}))$ can be written as the positive linear combination of vectors $γ_{j},$ $j \in J .$ Moreover, $〈 z - Π (z | L (V_{J})), γ_{j} 〉 = 0$ for $j \in J .$ From Lemma 1, there exists $J_{0} \subset J$ such that $V_{J_{0}}$ is linearly independent and $Π (z | L (V_{J}))$ can be written as a positive linear combination of the vectors in $V_{J_{0}},$ which implies that $Π (z | L (V_{J})) \in {\bar{F}}_{J_{0}} .$ In addition, since $〈 z - Π (z | L (V_{J})), γ_{j} 〉 = 0$ for $j \in J_{0},$ $Π (z | L (V_{J_{0}})) = Π (z | L (V_{J})) .$ Thus, $Π (z | Ω^{0}) = Π (z | L (V_{J_{0}})) .$ If $L (V_{J_{0}}) = L (V_{J})$ then $J^{*} = J_{0}$ satifies all required conditions. Now, assume that $L (V_{J_{0}}) \subset L (V_{J}) .$ The fact that $Π (z | L (V_{J_{0}})) = Π (z | L (V_{J}))$ implies that $Π (z | L (V_{J_{1}})) = Π (z | L (V_{J_{0}}))$ for any set $J_{1}$ such that $J_{0} \subseteq J_{1} \subseteq J .$ Further, since $Π (z | L (V_{J_{0}})) \in {\bar{F}}_{J_{0}}$ then $Π (z | L (V_{J_{1}})) \in {\bar{F}}_{J_{1}} .$ Thus, it is enough to choose the set $J^{*}$ such that $J_{0} \subset J^{*} \subset J$ and $V_{J^{*}}$ is a linearly independent set that spans $L (V_{J}) .$

Proof of Theorem 2. To prove this theorem, we start with a set $J \notin G_{μ}$ and find necessary conditions for such set to belong to $G_{s} .$ These necessary conditions, expressed as inequalities in terms of smooth and continuous functions of the ${\hat{N}}_{d} / N$ and the ${\hat{t}}_{d} / N,$ are then used to bound the probability of interest. Finally, we use Theorem 5.4.3 in Fuller (1996) to show that this probability converges to zero with a rate of $O (n^{- 1}) .$

Let $A_{μ},$ $A_{μ, J}$ and $γ_{μ_{d}}$ be the analogous versions of $A_{s},$ $A_{s, J}$ and $γ_{s_{d}}$ obtained by substituting ${\tilde{y}}_{s}$ and $W_{s}$ by $μ$ and $W_{μ},$ respectively. Lemma 2 ensures that both $A_{s}$ and $A_{μ}$ are irreducible since $A$ is.

First, suppose $\emptyset \notin G_{μ}$ and let $J = \emptyset .$ Then, from conditions in (2.8), $\emptyset \in G_{s}$ if and only if $〈 {\tilde{z}}_{s}, γ_{s_{j}} 〉 \leq 0$ for $j = 1, 2, \dots, m .$ In contrast, suppose that $〈 z_{μ}, γ_{μ_{j}} 〉 \leq 0$ for $j = 1, 2, \dots, m .$ Hence, $\emptyset \in G_{μ},$ which contradicts our choice of $J .$ Therefore, there exists $j_{0}$ such that $〈 z_{μ}, γ_{μ_{j_{0}}} 〉 > 0.$ Then, we have

$\begin{array}{l} P (\emptyset \in G_{s}) \leq P (0 \geq 〈 {\tilde{z}}_{s}, γ_{s_{j_{0}}} 〉) & = P (〈 z_{μ}, γ_{μ_{j_{0}}} 〉 - 〈 {\tilde{z}}_{s}, γ_{s_{j_{0}}} 〉 \geq 〈 z_{μ}, γ_{μ_{j_{0}}} 〉) \\ = P ({[\frac{〈 z_{μ}, γ_{μ_{j_{0}}} 〉 - 〈 {\tilde{z}}_{s}, γ_{s_{j_{0}}} 〉}{〈 z_{μ}, γ_{μ_{j_{0}}} 〉}]}^{2} \geq 1) \\ \leq \frac{1}{{〈 z_{μ}, γ_{μ_{j_{0}}} 〉}^{2}} E [{(〈 {\tilde{z}}_{s}, γ_{s_{j_{0}}} 〉 - 〈 z_{μ}, γ_{μ_{j_{0}}} 〉)}^{2}] \end{array}$

where the last inequality is obtained by an application of Markov’s inequality (see for example Casella and Berger (2002), Section 3.6.1). We show now that the expected value in the last term is $O (n^{- 1}) .$ Note that the expression inside of the expected value in the above inequality is a function of vector ${\hat{x}}_{s} = {(N^{- 1} {\hat{t}}_{1}, \dots, N^{- 1} {\hat{t}}_{D}, N^{- 1} {\hat{N}}_{1}, \dots, N^{- 1} {\hat{N}}_{D})}^{T} .$ Let $f_{1} (\cdot)$ be such a function (which does not depend on $N),$ and denote $x_{μ} = (r_{1} μ_{1}, \dots, r_{D} μ_{D}, r_{1}, \dots, r_{D}) .$ To apply Theorem 5.4.3 in Fuller (1996) with $α = 1, s = 2$ and $a_{N} = O (n^{- 1 / 2}),$ first we need to show that the following conditions are satisfied:

(a) $E [{({\hat{x}}_{s} - x_{μ})}^{2}] = O (n^{- 1}) .$

(b) $f_{1}$ is uniformly bounded in a closed and bounded sphere $S .$

$f_{1}^{(i_{1}, \dots, i_{r})} (x_{0}) = \frac{\partial^{r}}{\partial_{x_{i_{1}}} \dots \partial_{x_{i_{r}}}} {f_{1} (x) |}_{x = x_{0}} .$

(d) $x_{μ}$ is an interior point of $S .$

(e) There is a finite number $K$ such that

$| f_{1}^{(i_{1}, i_{2})} (x) | \leq K for all x \in S,$

$| f_{1}^{(i_{1})} (x_{μ}) | \leq K and | f_{1} (x_{μ}) | \leq K .$

Condition (a) is directly met by Lemma 4 (iv). In addition, Lemma 4 (i)-(ii) guarantees that there exist a constant $M > 1$ such that $| N^{- 1} {\hat{t}}_{d} | \leq M$ and $M^{- 1} \leq N^{- 1} {\hat{N}}_{d} \leq M .$ Hence, there exists a closed and bounded sphere $S$ that it is contained within these constant bounds. Moreover, from Assumption A3, we can conclude that $x_{μ} \in S,$ so condition (d) is satisfied. To show that condition (b) is met, note that $f_{1}$ is a continuous function in $S$ since both $W_{s}^{- 1 / 2}$ and ${\tilde{y}}_{s_{d}}$ exist for any $x \in S .$ Therefore, the Extreme Value Theorem (see Theorem 4.15 in Rudin (1976)) ensures that $f_{1}$ is uniformly bounded in $S .$ Conditions (c) and (e) are satisfied since $f_{1}$ is a continuous rational function in $S,$ implying that $f_{1}$ is infinitely differentiable and its derivatives are bounded in $S .$ Finally, all conditions (a)-(e) are fulfilled. Therefore, from Theorem 5.4.3 in Fuller (1996), we can conclude that $E [f_{1} (x)] = O (n^{- 1}),$ since $f_{1}$ and its first derivative with respect to the $N^{- 1} {\hat{t}}_{d}$ and $N^{- 1} {\hat{N}}_{d}$ evaluate to zero at $x_{μ} .$

Now, take any $J \neq \emptyset$ such that $J \notin G_{μ},$ and assume that $J \in G_{s} .$ Theorem 1 guarantees that we can always choose a subset $J^{*} \subseteq J$ such that $J^{*} \in G_{s},$ $V_{s, J^{*}}$ is linearly independent, and $L (V_{s, J^{*}}) = L (V_{s, J}) .$ Note that $Π ({\tilde{z}}_{s} | L (V_{s, J^{*}})) = A_{s, J^{*}}^{T} {(A_{s, J^{*}} A_{s, J^{*}}^{T})}^{- 1} A_{s, J^{*}} {\tilde{z}}_{s} .$ Let ${\tilde{b}}_{s, J^{*}} = {(A_{s, J^{*}} A_{s, J^{*}}^{T})}^{- 1} A_{s, J^{*}} {\tilde{z}}_{s} .$ Hence, from conditions in (2.8), we have that $J \in G_{s}$ implies that ${\tilde{b}}_{s, J^{*}} \geq 0,$ and $〈 {\tilde{z}}_{s} - A_{s, J^{*}}^{T} {\tilde{b}}_{s, J^{*}}, γ_{s_{j}} 〉 \leq 0$ for any $j .$ Define $b_{μ, J^{*}} = {(A_{μ, J^{*}} A_{μ, J^{*}}^{T})}^{- 1} A_{μ, J^{*}} z_{μ}$ and assume that $b_{μ, J^{*}} \geq 0,$ and $〈 z_{μ} - A_{μ, J^{*}}^{T} b_{μ, J^{*}}, γ_{μ_{j}} 〉 \leq 0$ for $j = 1, 2, \dots, m .$ These conditions would imply that $J^{*} \in G_{μ},$ contradicting the original assumption that $J \notin G_{μ},$ since $L (V_{μ, J^{*}}) = L (V_{μ, J})$ from Lemma 3. Therefore, either there is an element of $b_{μ, J^{*}}$ that is strictly negative or there exists $j_{0}$ such that $〈 z_{μ} - A_{μ, J^{*}}^{T} b_{μ, J^{*}}, γ_{μ_{j_{0}}} 〉 > 0.$ Hence, proving that $P (J \in G_{s}) = O (n^{- 1})$ in any of these two scenarios will conclude the proof.

Suppose the $j_{0}^{th}$ element of $b_{μ, J^{*}}$ is strictly negative. That is, $e_{j_{0}}^{T} b_{μ, J^{*}} < 0,$ where $e_{j}$ denotes the indicator vector that is 1 for entry $j$ and 0 otherwise. Then, we have

$\begin{array}{l} P (J \in G_{s}) \leq P (e_{j_{0}}^{T} {\tilde{b}}_{s, J^{*}} \geq 0) & = P (e_{j_{0}}^{T} {\tilde{b}}_{s, J^{*}} - e_{j_{0}}^{T} b_{μ, J^{*}} \geq - e_{j_{0}}^{T} b_{μ, J^{*}}) \\ \leq \frac{1}{{(e_{j_{0}}^{T} b_{μ, J^{*}})}^{2}} E [{(e_{j_{0}}^{T} {\tilde{b}}_{s, J^{*}} - e_{j_{0}}^{T} b_{μ, J^{*}})}^{2}] . \end{array}$

Denote $f_{2} ({\hat{x}}_{s})$ to the expression inside the above expected value. An analogous argument to the one used for the function $f_{1}$ can be applied to the rational continuous function $f_{2}$ over $S,$ to conclude that $E [f_{2} ({\hat{x}}_{s})] = O (n^{- 1}) .$ Note that we also used the fact that $A_{s, J^{*}} A_{s, J^{*}}^{T}$ is an invertible matrix for any $x \in S .$

Lastly, suppose there exists $j_{0}$ such that $κ_{z_{μ}, j_{0}} = 〈 z_{μ} - A_{μ, J^{*}}^{T} b_{μ, J^{*}}, γ_{μ_{j_{0}}} 〉 > 0,$ and denote $κ_{{\tilde{z}}_{μ}, j_{0}} = 〈 {\tilde{z}}_{s} - A_{s, J^{*}}^{T} {\tilde{b}}_{s, J^{*}}, γ_{s_{j_{0}}} 〉 .$ Then, we have

$\begin{array}{l} P (J \in {\tilde{G}}_{s}) \leq P (0 \geq κ_{{\tilde{z}}_{s}, j_{0}}) & = P (κ_{z_{μ}, j_{0}} - κ_{{\tilde{z}}_{s}, j_{0}} \geq κ_{z_{μ}, j_{0}}) \\ \leq \frac{1}{κ_{z_{μ}, j_{0}}^{2}} E [{(κ_{{\tilde{z}}_{s}, j_{0}} - κ_{z_{μ}, j_{0}})}^{2}] . \end{array}$

Denote $f_{3} ({\hat{x}}_{s})$ to the expression inside the above expected value. An analogous argument to the one used for the functions $f_{1}, f_{2}$ is applied to conclude that $E [f_{3} ({\hat{x}}_{s})] = O (n^{- 1}) .$

Proof of Theorem 3. Take any $J \in G_{s}$ and any domain $d .$ Note that the condition $A μ \geq 0$ implies that $\emptyset \in G_{μ} .$ Then, we can write ${\tilde{θ}}_{s_{d}} - {\bar{y}}_{U_{d}}$ as

${\tilde{θ}}_{s_{d}} - {\bar{y}}_{U_{d}} = ({\tilde{y}}_{s_{d}} - {\bar{y}}_{U_{d}}) 1_{J = \emptyset} + \sum_{J_{G} \in G_{μ} \ \emptyset} ({\tilde{θ}}_{s_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J_{G} = J} + \sum_{J_{G} \in G_{μ}^{c}} ({\tilde{θ}}_{s_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J_{G} = J},$

where we used that ${\tilde{θ}}_{s_{d}, \emptyset} = {\tilde{y}}_{s_{d}} .$ Now, an unfeasible variance estimator $AV ({\tilde{θ}}_{s_{d}, J})$ can be written as

$AV ({\tilde{θ}}_{s_{d}, J}) = AV ({\tilde{y}}_{s_{d}}) 1_{J = \emptyset} + \sum_{J_{G} \in G_{μ} \ \emptyset} AV ({\tilde{θ}}_{s_{d}, J_{G}}) 1_{J = J_{G}} + \sum_{J_{G} \in G_{μ}^{c}} AV ({\tilde{θ}}_{s_{d}, J_{G}}) 1_{J = J_{G}} .$

Hence,

$\begin{array}{l} AV {({\tilde{θ}}_{s_{d}, J})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}} - {\bar{y}}_{U_{d}}) & = AV {({\tilde{y}}_{s_{d}})}^{- 1 / 2} ({\tilde{y}}_{s_{d}} - {\bar{y}}_{U_{d}}) 1_{J = \emptyset} + \sum_{J_{G} \in G_{μ} \ \emptyset} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J = J_{G}} + \sum_{J_{G} \in G_{μ}^{c}} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J = J_{G}} \\ = [AV {({\tilde{y}}_{s_{d}})}^{- 1 / 2} ({\tilde{y}}_{s} - {\bar{y}}_{U_{d}}) 1_{J = \emptyset} + \sum_{J_{G} \in G_{μ} \ \emptyset} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}, J_{G}} - θ_{U_{d}, J_{G}}) 1_{J = J_{G}} + \sum_{J_{G} \in G_{μ}^{c}} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}, J_{G}} - θ_{U_{d}, J_{G}}) 1_{J = J_{G}}] \\ + [\sum_{J_{G} \in G_{μ} \ \emptyset} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} (θ_{U_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J = J_{G}}] + [\sum_{J_{G} \in G_{μ}^{c}} AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} (θ_{U_{d}, J_{G}} - {\bar{y}}_{U_{d}}) 1_{J = J_{G}}] \\ = c_{1 N} + c_{2 N} + c_{3 N}, \end{array}$

where $θ_{U_{d}, J_{G}}$ is the population version of ${\tilde{θ}}_{s_{d}, J_{G}} .$ A first order term Taylor expansion of ${\tilde{θ}}_{s_{d}, J_{G}}$ and Assumption A6 allow to conclude that each term of the form

$AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}, J_{G}} - θ_{U_{d}, J_{G}})$

converges in distribution to a standard normal distribution. Therefore, $c_{1 N}$ also converges to a standard normal distribution. Note that for each $J_{G} \in G_{μ}^{c},$

$AV {({\tilde{θ}}_{s_{d}, J_{G}})}^{- 1 / 2} (θ_{U_{d}, J_{G}} - {\bar{y}}_{U_{d}}) = {[n AV ({\tilde{θ}}_{s_{d}, J_{G}})]}^{- 1 / 2} [n^{1 / 2} (θ_{U_{d}, J_{G}} - {\bar{y}}_{U_{d}})] = O (n^{1 / 2}),$

while $1_{J = J_{G}} = O_{p} (n^{- 1})$ by Theorem 2 (since $J \in G_{s}) .$ Thus, $c_{3 N} = O_{p} (n^{- 1 / 2}) .$ Now, note that $θ_{U_{d}, J_{G}} - {\bar{y}}_{U_{d}} = O (N^{- 1 / 2})$ when $J_{G} \in G_{μ} \ \emptyset$ by Assumption A3. Hence, for any $J_{G} \in G_{μ} \ \emptyset,$

which implies that $c_{2 N} = O (\sqrt{\frac{n}{N}})$ (bias term). Thus, by combining these properties of $c_{1 N},$ $c_{2 N}$ and $c_{3 N},$ we conclude that

$AV {({\tilde{θ}}_{s_{d}, J})}^{- 1 / 2} ({\tilde{θ}}_{s_{d}} - {\bar{y}}_{U_{d}}) \overset{L}{\to} N (B, 1),$

where $B = O (\sqrt{\frac{n}{N}}) .$

Now, write the feasible variance estimator $\hat{V} ({\tilde{θ}}_{s_{d}, J})$ as

$\hat{V} ({\tilde{θ}}_{s_{d}, J}) = \hat{V} ({\tilde{y}}_{s_{d}}) 1_{J = \emptyset} + \sum_{J_{G} \in G_{μ} \ \emptyset} \hat{V} ({\tilde{θ}}_{s_{d}, J_{G}}) 1_{J = J_{G}} + \sum_{J_{G} \in G_{μ}^{c}} \hat{V} ({\tilde{θ}}_{s_{d}, J_{G}}) 1_{J = J_{G}} .$

By Assumption A6, we have that $\hat{V} ({\tilde{θ}}_{s_{d}, J_{G}}) - AV ({\tilde{θ}}_{s_{d}, J_{G}}) = o_{p} (n^{- 1})$ for any $J_{G},$ which implies that $\hat{V} {({\tilde{θ}}_{s_{d}, J})}^{1 / 2} - AV {({\tilde{θ}}_{s_{d}, J})}^{1 / 2} = o_{p} (n^{- 1 / 2}) .$ Hence, an application of Slutsky’s theorem allows to replace $AV {({\tilde{θ}}_{s_{d}, J})}^{- 1 / 2}$ by $\hat{V} {({\tilde{θ}}_{s_{d}, J})}^{- 1 / 2} .$

To prove the last part of this theorem, just note that $A μ > 0$ implies $G_{μ} = {\emptyset} .$ Thus, the term $c_{2 N}$ does not exist and the bias term vanishes.

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Estimation and inference of domain means subject to qualitative constraints
Section 6. Conclusions

Appendix

References

Estimation and inference of domain means subject to qualitative constraints Section 6. Conclusions

Appendix

References

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Estimation and inference of domain means subject to qualitative constraints
Section 6. Conclusions