Estimation and inference of domain means subject to qualitative constraints
Section 3. Properties of the constrained estimator

• Previous
• Next

3.1  Assumptions

To derive our theoretical results, we make assumptions on the asymptotic behavior of the population $U_N$ and the sampling design $p_N:$

$$\underset{N\to \infty }{{\displaystyle \lim sup}}{n}_N\underset{k\mathrm{,}l\in U_N\text{}\mathrm{<mo>:</mo>}k\ne l}{{\displaystyle \max }}\left|{\Delta }_{kl}\right|\mathrm{<mo><</mo>}\infty \mathrm{.}$$

$${\text{var}}_{p_N}{\left({\widehat{x}}_{s_N}\right)}^{-1/2}\left({\widehat{x}}_{s_N}-{\overline{x}}_{U_N}\right)\stackrel{d}{{\displaystyle \to }}N\left(0\mathrm{,}{I}_{2D}\right)\mathrm{,}$$

$$\widehat{\mathrm{var}}\left({\widehat{x}}_{s_N}\right)-{\text{var}}_{p_N}\left({\widehat{x}}_{s_N}\right)\mathrm{<mo>=</mo>}{o}_p\left(n_N^{-1}\right)\mathrm{<mo>;</mo>}$$

Assumption A1 establishes that the number of domains remains constant as the population size changes. The condition in Assumption A2 is made to ensure design consistency of Horvitz-Thompson estimators at the population and domain levels. In particular, note that this condition is satisfied when the variable $y$ is bounded, which can be naturally assumed for many types of survey variables. Assumption A3 guarantees that the population domain means and sizes converge to the limiting values ${\mu }_d$ and $r_d,$ respectively. Alternatively, the $\mu$ values can be thought as superpopulation expectations for a distribution that generates the population elements $y_k$ as independent draws. In fact, our theoretical results depend on whether the assumed constraints hold for these superpopulation expectations and not for the population domain means. Although this might seem to be inappropriate given our interest on using constraints at the population level, Assumption A3 ensures that the shape of the domain means would be reasonably close to the shape of the superpopulation means. Assumption A4 states that the sample size in each domain cannot be smaller than a fraction of the ratio $n/D,$ which would be obtained by dividing equally the sample size over all domains. This assumption aims to ensure that the moments of smooth functions of the $N^{-1}{\widehat{t}}_d$ and the $N^{-1}{\widehat{N}}_d$ are bounded. Also, it assumes that the sample size is non-random. This can be adapted to a random sample size by imposing certain conditions on the expected sample size $E_p\left(n\right).$ Assumption A5 establishes non-zero lower bounds for both first and second order inclusion probabilities, and states that the design covariances ${\Delta }_{kl}$ must converge to zero at least as fast as $n^{-1}.$ Assumption A6 ensures asymptotic normality for ${\widehat{x}}_{s_N},$ which is needed to maintain normality properties on non-linear estimators that are expressed as smooth functions of ${\widehat{x}}_{s_N}.$ It is also used to establish consistency conditions on the variance-covariance estimator. For specific designs, asymptotic normality results are available in the literature, including the classical result by Hájek (1960) for Poisson sampling and simple random sampling without replacement. Additional central limit theorems for stratified sampling include Krewski and Rao (1981), who considered stratified unequal probability samples with replacement, Bickel and Freedman (1984), who considered stratified simple random sampling without replacement, and Breidt, Opsomer and Sanchez-Borrego (2016), who considered general unequal probability designs, with or without replacement.

3.2  Main results

We derive the theoretical properties of the constrained estimator by focusing on the projection onto ${\Omega }_s^0$ instead of ${\Omega }_s.$ Recall that the edges of the polar cone ${\Omega }_s^0$ are simply the $m$ rows of $-A_s,$ denoted by ${\gamma }_{s_j};$ and that ${\tilde{\rho }}_s,$ the projection onto ${\Omega }_s^0,$ can be described by the sets $J\in G_s.$ Being able to characterize the property that $J\in G_s$ in terms of the vectors in $V_{s\mathrm{,}J}$ allow us to obtain theoretical convergence rates, which are used to develop inference properties of the constrained estimator. When the set $J\in G_s$ produces a set of linear independent vectors $V_{s\mathrm{,}J},$ then it is straightforward that ${\tilde{\rho }}_s$ can be written as $P_{s\mathrm{,}J}{\tilde{z}}_s\mathrm{<mo>=</mo>}{A}_{s\mathrm{,}J}^T{\left(A_{s\mathrm{,}J}{A}_{s\mathrm{,}J}^T\right)}^{-1}{A}_{s\mathrm{,}J}{\tilde{z}}_s,$ where $A_{s\mathrm{,}J}$ denotes the matrix formed by the rows of $A_s$ in positions $J.$ Hence, based on the conditions in (2.8), $J\in G_s$ if and only if

$$\langle {\tilde{z}}_s-P_{s\mathrm{,}J}{\tilde{z}}_s\mathrm{,}{\gamma }_{s_j}\rangle \le 0\text{for}j\notin J\mathrm{,}\text{and}{\left(A_{s\mathrm{,}J}{A}_{s\mathrm{,}J}^T\right)}^{-1}{A}_{s\mathrm{,}J}{\tilde{z}}_s\ge 0(3.1)$$

in this case, where the latter condition assures that $\Pi \left({\tilde{z}}_s|L\left(V_{s\mathrm{,}J}\right)\right)\in {\overline{F}}_{s\mathrm{,}J}.$ However, it is possible that the set $J\in G_s$ produces a set of linearly dependent vectors $V_{s\mathrm{,}J}.$ In that case, Theorem 1 below guarantees that it is always possible to find a subset $J^{*}\subset J$ such that $V_{s\mathrm{,}{J}^{*}}$ is a linearly independent set that spans the same linear space as $V_{s\mathrm{,}J}$ and that satisfies $J^{*}\in G_s.$ Thus, analogous conditions as in (3.1) can be established using $J^{*}$ instead of $J.$

Theorem 1. Let $A$  be a $m\times D$  irreducible matrix with rows $-{\gamma }_j.$  Let ${\Omega }^0$  be its corresponding polar cone. For any set $J\subseteq \left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\},$  define $V_J\mathrm{<mo>=</mo>}\left\{{\gamma }_j\text{}\mathrm{<mo>:</mo>}j\in J\right\}.$  Further, denote ${\overline{F}}_J$  to be the subcone of ${\Omega }^0$  generated by the edges given by the set $J.$  For a vector $z,$  define its set $G$  to be formed by all sets $J\subseteq \left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\}$  such that $\Pi \left(z|{\Omega }^0\right)\mathrm{<mo>=</mo>}\Pi \left(z|L\left(V_J\right)\right)\in {\overline{F}}_J.$  Suppose $J$  is a non-empty set such that $V_J$  is a linearly dependent set and $J\in G.$ Then, there exists $J^{*}\subset J$  such that $V_{J^{*}}$  is a linearly independent set, $L\left(V_{J^{*}}\right)\mathrm{<mo>=</mo>}L\left(V_J\right),$  and $J^{*}\in G.$

All above concepts that have been defined at the sample level can be analogously defined at the superpopulation level. In particular, let $G_{\mu }$ be the set of all subsets $J\subseteq \left\{\mathrm{1,}\dots \mathrm{,}m\right\}$ such that $\Pi \left(z_{\mu }|{\Omega }_{\mu }^0\right)\mathrm{<mo>=</mo>}\Pi \left(z_{\mu }|L\left(V_{\mu \mathrm{,}J}\right)\right)\in {\overline{F}}_{\mu \mathrm{,}J},$ where $z_{\mu },$ ${\Omega }_{\mu }^0,$ $V_{\mu \mathrm{,}J}$ and ${\overline{F}}_{\mu \mathrm{,}J}$ are the analogous versions of ${\tilde{z}}_s,$ ${\Omega }_s^0,$ $V_{s\mathrm{,}J}$ and ${\overline{F}}_{s\mathrm{,}J}$ obtained by substituting ${\tilde{y}}_s$ and $W_s$ by $\mu \mathrm{<mo>=</mo>}\left({\mu }_1\mathrm{,}\dots \mathrm{,}{\mu }_D\right)$ and $W_{\mu }\mathrm{<mo>=</mo>}\text{diag}\left(r_1\mathrm{,}{r}_2\mathrm{,}\dots \mathrm{,}{r}_D\right).$ Necessary and sufficient conditions as in (2.8) can be analogously established to characterize the vector ${\rho }_{\mu }$ to be the projection onto ${\Omega }_{\mu }^0.$

Recall the set $G_s$ could vary for different samples. Also, note that highly variable small samples are likely to choose sets $J\in G_s$ that are not chosen in the “asymptotically correct” $G_{\mu }.$ However, as the sample size increases, these incorrect choices are less likely to occur since the sample domain means get closer to the limiting population domain means. This idea is made more precise in Theorem 2, which states that sets that are not in $G_{\mu }$ have an asymptotically negligible probability of being chosen in the sample.

Theorem 2. Consider any set $J\subseteq \left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}m\right\}$  such that $J\notin G_{\mu }.$ Then, $P\left(J\in G_s\right)\mathrm{<mo>=</mo>}O\left(n^{-1}\right).$

Theorem 3 below shows the asymptotic normality of the constrained estimator and justifies the use of the linearization-based variance estimator for the observed projection (or pooling, in the case of partial ordering) for asymptotic inference for the finite population domain mean. This generalizes Theorem 2 of Wu et al. (2016), where only monotone restrictions were considered. Note the presence of a bias term $B$ in the mean of the asymptotic distribution. This undesirable situation occurs when there is more than one set $J\in G_{\mu }$ such that their corresponding edges in $V_{\mu \mathrm{,}J}$ span different linear spaces, or equivalently, that the projection onto the polar cone ${\Omega }_{\mu }^0$ belongs to the intersection of those different linear spaces. However, when the constraints hold strictly, i.e., $A\mu \mathrm{<mo>></mo>}0,$ the vector $z_{\mu }$ is strictly inside the constraint cone ${\Omega }_{\mu },$ and in this case there is no set $J\ne \varnothing$ such that $\Pi \left(z_{\mu }|L\left(V_{\mu \mathrm{,}J}\right)\right)\mathrm{<mo>=</mo>}0.$ Thus, in this case, the bias term vanishes.

Theorem 3. Suppose that $\mu$  satisfies $A\mu \ge 0.$  Consider any set $J$  such that $J\in G_s.$ Then

$$\widehat{V}{\left({\tilde{\theta }}_{s_d\mathrm{,}J}\right)}^{-1/2}\left({\tilde{\theta }}_{s_d}-{\overline{y}}_{U_d}\right)\stackrel{L}{{\displaystyle \to }}N\left(B\mathrm{,}1\right)\mathrm{,}$$

for any $d\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}D,$  where $B\mathrm{<mo>=</mo>}O\left(\sqrt{\frac{n}{N}}\right)$  is a bias term that vanishes when $A\mu \mathrm{<mo>></mo>}0.$

Theorem 3 relies on the fact that the assumed shape constraints hold for the vector of limiting domain means $\mu$ instead of for the vector of population domain means ${\overline{y}}_U.$ In the next section, we show through simulations that the constrained estimator improves both estimation and variability when the population domains are approximately close to the assumed shape, in comparison with unconstrained estimators.

• Previous
• Article main page
• Next