2 Consistency and asymptotic normality

Jun Shao, Eric Slud, Yang Cheng, Sheng Wang, and Carma Hogue

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To consider asymptotics, we view the population $U$ as one of a sequence of populations $\left\{U^{\left(m\right)}\mathrm{,}m=1,2,\dots \right\},$ where the number of units in $U^{\left(m\right)}$ increases to infinity as $m\to \infty .$ This paper treats only the case of strata in which a large sample $n_h$ is drawn; that is, we assume that for each stratum $h,$ the sample size $n_h$ depends on $m$ and increases to infinity as $m\to \infty ,$ but we omit the index $m$ for simplicity. All limiting processes are considered as $m\to \infty .$Following authors such as Isaki and Fuller (1982) and Deville and Särndal (1992), we term this a superpopulation asymptotic framework. Under the design-based framework considered in Section 2.1, the attribute vectors in the underlying populations need not be viewed as random vectors. However, under the model-assisted framework considered in Section 2.2, regression models are assumed for attribute vectors.

Since each estimator is a sum of independent estimators constructed within each stratum, for simplicity we present asymptotic results for the case of $H=1.$ The results and conclusions immediately apply to the case of a fixed $H$ and can also be extended to the situation where $H$ increases to infinity. (It is typical for large-scale surveys to have many strata, although the number of ASPEP government-by-type strata that were split into substrata was somewhat less than 100.) Since we only consider $H=\mathrm{1,}$ we omit the index $h$ for stratum in this section, e.g., $n_{hj}=n_j,$ $n_h=n,$ $N_{hj}=N_j,$ and $N_h=N.$ Also, for $j=1,2,$ the estimators ${\widehat{\beta }}_j$ and $\widehat{\beta }$ are defined by the displayed formulas following equations (1.2) and (1.3), with subscript $h$ suppressed, together with

$$\begin{array}{lll}{\widehat{\mu }}_{xj}\hfill & =\hfill & {\widehat{X}}_j/{\widehat{N}}_j,{\widehat{\alpha }}_j={\widehat{Y}}_j/{\widehat{N}}_j-{\widehat{\beta }}_j{\widehat{\mu }}_{xj}\mathrm{,}{\widehat{\sigma }}_{xj}^2={\widehat{N}}_j^{-1}{\displaystyle \sum _{i\in S_j}}{\pi }_i^{-1}{\left(x_i-{\widehat{\mu }}_{xj}\right)}^2\hfill \\ {\widehat{\sigma }}_{xe\mathrm{,}j}^2\hfill & =\hfill & n_j{\displaystyle \sum _{i\in S_j}}{\left(x_i-{\widehat{\mu }}_{xj}\right)}^2{\left(y_i-{\widehat{\alpha }}_j-{\widehat{\beta }}_j{x}_i\right)}^2/\left({\pi }_i^2{\widehat{N}}_j^2\right).\hfill \end{array}$$

Furthermore, for simplicity we consider asymptotics only under with-replacement sampling. The results can be applied to the case of without replacement sampling if the sampling fraction $n/N$ is negligible.

2.1 Design-based asymptotic framework

First, we establish the asymptotic normality of ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ and ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ under repeated sampling, that is, when $y_i$ and $x_i$ are fixed for $i\in U,$ and $S_j$ is a random PPS sample.

Theorem 1 Suppose that $S_1$ and $S_2$ are independent PPS samples with replacement from $U_1$ and $U_2,$ respectively, where unit $i\in U_j$ has probability $p_{ij}=z_i/{\displaystyle {\sum }_{i\in U_j}}{z}_i\text{>}0$ of being selected, and sampling weight ${\pi }_i^{-1}=1/\left(n_j{p}_{ij}\right)$ for $j=1,2,$ and that the following four conditions hold, as the population sequence index $m$ goes to $\infty .$

(C1) There exist constants ${\phi }_j$ and ${\omega }_j$ such that $\sqrt{n/n_j}\to {\phi }_j$ and $N_j/N\to {\omega }_j.$

(C2)  For $j=1,2,$ there exist constants ${\mu }_{yj}\mathrm{,}{\mu }_{xj}$ and ${\beta }_j$ such that

$${\overline{Y}}_j=Y_j/N_j={\displaystyle \sum _{i\in U_j}}{y}_i/N_j\to {\mu }_{yj}\mathrm{,}{\overline{X}}_j=X_j/N_j={\displaystyle \sum _{i\in U_j}}{x}_i/N_j\to {\mu }_{xj}$$

exist, as do the limits $N_j^{-1}{\displaystyle {\sum }_{i\in U_j}}{\left(x_i-{\mu }_{xj}\right)}^2\to {\sigma }_{xj}^2\text{>}0,$ and in addition,

$$\left(\sqrt{n_j}/N_j\right){\displaystyle \sum _{i\in U_j}}{x}_i\left(y_i-Y_j/N_j-{\beta }_j\left(x_i-X_j/N_j\right)\right)\to 0\text{as}n\mathrm{,}N\to \infty \mathrm{.}$$

(C3) The limits $D_{N_j}={\displaystyle {\sum }_{i\in U_j}}{p}_{ij}{b}_{ij}{b}_{ij}^T/N_j^2\to D_j$ exist, where for $i\in U_j,$

$$b_{ij}={\left[1/p_{ij}-N_j\mathrm{,}{x}_i/p_{ij}-X_j\mathrm{,}{y}_i/p_{ij}-Y_j\right]}^T\mathrm{,}$$

$v^T$ denotes the vector transpose, and $D_j$ is positive definite. The limit ${\sigma }_{xe\mathrm{,}j}^2=$ $\lim N_j^{-2}{\displaystyle {\sum }_{i\in U_j}}{\left(x_i-{\mu }_{xj}\right)}^2{\left(y_i-{\alpha }_j-{\beta }_j{x}_i\right)}^2/p_{ij}$ also exists, for ${\alpha }_j={\mu }_{yj}-{\beta }_j{\mu }_{xj}\mathrm{.}$

(C4) The elements of ${\Lambda }_j={\displaystyle {\sum }_{i\in U_j}}{p}_{ij}{c}_{ij}{c}_{ij}^T/N_j^4$ form a bounded sequence, where for $i\in U_j,$

$$c_{ij}={\left[{\left(1/p_{ij}-N_j\right)}^2\mathrm{,}{\left(x_i/p_{ij}-X_j\right)}^2\mathrm{,}{\left(y_i/p_{ij}-Y_j\right)}^2\right]}^T.$$

Then, as $m\to \infty ,$ the following conclusions hold.

(a) For $j=1,2,$ ${\widehat{\mu }}_{xj}{\to }_{{}_P}{\mu }_{xj}\mathrm{,}{\widehat{\mu }}_{yj}{\to }_{{}_P}{\mu }_{yj}\mathrm{,}{\widehat{\beta }}_j{\to }_{{}_P}{\beta }_j\mathrm{,}{\widehat{\alpha }}_j{\to }_{{}_P}{\alpha }_j,$ and ${\widehat{\sigma }}_{xj}^2{\to }_{{}_P}{\sigma }_{xj}^2,$ where ${\to }_{{}_P}$ denotes convergence in probability.

(b) The combined-stratum estimator $\widehat{\beta }$ has the exact expression

$$\widehat{\beta }=\frac{{\displaystyle {\sum }_{j=1}^2{\widehat{\beta }}_j{\widehat{\sigma }}_{xj}^2{\widehat{N}}_j}+\left({\widehat{X}}_2-{\widehat{X}}_1\right)\left({\widehat{Y}}_2-{\widehat{Y}}_1\right){\widehat{N}}_1{\widehat{N}}_2/\left({\widehat{N}}_1+{\widehat{N}}_2\right)}{{\displaystyle {\sum }_{j=1}^2{\widehat{\sigma }}_{xj}^2{\widehat{N}}_j}+{\left({\widehat{X}}_2-{\widehat{X}}_1\right)}^2{\widehat{N}}_1{\widehat{N}}_2/\left({\widehat{N}}_1+{\widehat{N}}_2\right)} (2.1)$$

and the in-probability limit

$$\beta =\frac{{\displaystyle {\sum }_{j=1}^2{\beta }_j{\sigma }_{xj}^2{\omega }_j}+\left({\mu }_{x2}-{\mu }_{x1}\right)\left({\mu }_{y2}-{\mu }_{y1}\right){\omega }_1{\omega }_2}{{\displaystyle {\sum }_{j=1}^2{\sigma }_{xj}^2{\omega }_j}+{\left({\mu }_{x2}-{\mu }_{x1}\right)}^2{\omega }_1{\omega }_2}\mathrm{.}$$

(c)  $\sqrt{n_j}\left({\widehat{\beta }}_j-{\beta }_j\right){\to }_{d}N\left(\mathrm{0,}{\sigma }_{xe\mathrm{,}j}^2/{\sigma }_{x\mathrm{,}j}^4\right),$ where ${\to }_d$ denotes convergence in distribution, and ${\widehat{\sigma }}_{xe\mathrm{,}j}^2{\to }_{{}_P}{\sigma }_{xe\mathrm{,}j}^2.$

(d) For $k=1,2,$

$$\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,}k}-Y\right)/N{\to }_{d}N\left(\mathrm{0,}{\sigma }_k^2\right) (2.2)$$

where ${\sigma }_k^2={\displaystyle {\sum }_{j=1}^2}{a}_{kj}^T{D}_j{a}_{kj}$ and

$$a_{1j}={\omega }_j{\phi }_j{\left[-\left({\mu }_y-\beta {\mu }_x\right)\mathrm{,}-\beta \mathrm{,1}\right]}^T\mathrm{,}{a}_{2j}={\omega }_j{\phi }_j{\left[-\left({\mu }_{yj}-{\beta }_j{\mu }_{xj}\right)\mathrm{,}-{\beta }_j\mathrm{,1}\right]}^T\mathrm{,}$$

${\mu }_x={\omega }_1{\mu }_{x1}+{\omega }_2{\mu }_{x2},$ ${\mu }_y={\omega }_1{\mu }_{y1}+{\omega }_2{\mu }_{y2},$ and $D_j$ is given in condition (C3).

The conditions (C1)-(C4) of Theorem 1 provide a general formulation of the superpopulation framework for large-sample design-based statistical inference, within which the survey regression coefficients estimate well-defined frame-population descriptive parameters. The results in parts (a)-(b) show that the in-probability limits ${\beta }_j\mathrm{,}{\alpha }_j$ of ${\widehat{\beta }}_j\mathrm{,}{\widehat{\alpha }}_j$ have the standard interpretation as superpopulation least-squares slopes and intercepts. (These slope and intercept parameters also keep their usual model-based interpretations under the model (2.7) introduced in Section 2.2.) The asymptotic distribution theory for ${\widehat{\beta }}_j$ in conclusion (c) allows us to deduce the large-sample behavior of ${\widehat{Y}}_{\text{dec}}$ from that provided in (d) for ${\widehat{Y}}_{\text{reg}\mathrm{,}k}.$

Under the further conditions

$${\beta }_1={\beta }_2\mathrm{,}{\alpha }_1={\alpha }_2\mathrm{,} (2.3)$$

it is clear from Theorem 1(b) that ${\beta }_j=\beta ,$ and ${\sigma }_1^2={\sigma }_2^2$ in (2.2), so that ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ and ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ and ${\widehat{Y}}_{\text{dec}}$ are all asymptotically the same up to remainders of smaller order than $N/\sqrt{n},$ as we now show. Also, if ${\beta }_1\ne {\beta }_2,$ then ${\widehat{Y}}_{\text{reg}\mathrm{,2}}-{\widehat{Y}}_{\text{dec}}$ continues to be $o_P\left(N/\sqrt{n}\right),$ and the test of equality of slopes rejects, i.e., $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)\to 1,$ and therefore ${\widehat{Y}}_{\text{dec}}$ has the same asymptotic distribution as ${\widehat{Y}}_{\text{reg}\mathrm{,2}},$ which is more efficient than ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ according to the result in Section 2.2.

Theorem 2 Assume the same hypotheses (C1)-(C4) as in Theorem 1.

(a) When (2.3) holds, then as $m\to \infty$

$$\sqrt{n}\left({\widehat{\beta }}_2-{\widehat{\beta }}_1\right){\to }_{d}N\left(\mathrm{0,}{\sigma }_d^2\right)\mathrm{,}{\sigma }_d^2={\displaystyle \sum _{j=1}^2}\frac{{\sigma }_{xe\mathrm{,}j}^2}{{\phi }_j^2{\sigma }_{xj}^4}\mathrm{,} (2.4)$$

and the estimators ${\widehat{Y}}_{\text{reg}\mathrm{,1}}\mathrm{,}{\widehat{Y}}_{\text{reg}\mathrm{,2}},$ are all asymptotically normally distributed and equivalent in the sense that

$$\frac{n}{N^2}\left[{\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-{\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)}^2+{\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-{\widehat{Y}}_{\text{dec}}\right)}^2\right]{\to }_{{}_P}0 (2.5)$$

(b) When ${\beta }_1\ne {\beta }_2,$ $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)\to 1$ and $\sqrt{n}\left({\widehat{Y}}_{\text{dec}}-Y\right)/N{\to }_{{}_d}N\left(\mathrm{0,}{\sigma }_2^2\right).$

A more refined study of the asymptotic behavior of the estimators ${\widehat{Y}}_{\text{dec}}$ can be undertaken in the spirit of Saleh (2006), as with contiguous or Pitman alternatives for non-survey statistical models, by assuming that $\sqrt{n}\left({\beta }_1-{\beta }_2\right)\to r$ for a constant $r.$ Under this assumption, it can be shown that ${\widehat{Y}}_{\text{reg}\mathrm{,1}}-{\widehat{Y}}_{\text{reg}\mathrm{,2}}=o_P\left(N/\sqrt{n}\right)$ and, therefore, the three centered and scaled estimators $\sqrt{n}\left({\widehat{Y}}_{\text{dec}}-Y\right)\mathrm{,}$ $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right),$ and $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-Y\right)$all have the same asymptotic normal distribution with mean 0. Furthermore,

$$P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)\to \Phi \left(-z_{\tau /2}+r/{\sigma }_d\right)+\Phi \left(-z_{\tau /2}-r/{\sigma }_d\right)\mathrm{,} (2.6)$$

where ${\sigma }_d^2$ is given in (2.4), and $z_{\tau /2}$ and $\Phi$ are respectively the standard normal percentage point and distribution function. Thus, $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right)$ has a limit different from 1. In particular, the limit in (2.6) equals $\tau$ when ${\beta }_1={\beta }_2$ (i.e., when $r=0$ ).

2.2 Model-assisted asymptotic setting

We elaborate in this section the behavior of estimators ${\widehat{Y}}_{\text{reg}\mathrm{,}k}\mathrm{,}{\widehat{Y}}_{\text{dec}}$ under the assumed probabilistic model that the triples $(x_i\mathrm{,}{y}_i\mathrm{,}{z}_i)$ in the finite population, $i\in U_j,$ are independent and identically distributed (iid), where the size-variables $z_i\text{>}0$ are used in defining PPS with-replacement draw probabilities $p_{ij}=z_i/{\displaystyle {\sum }_{i^{\prime }\in U_j}}{z}_{i^{\prime }},$ and where $x_i$ and $y_i$ follow the model

$$y_i={\alpha }_j+{\beta }_j{x}_i+{\epsilon }_i\mathrm{,}i\in U_j\mathrm{,} (2.7)$$

with ${\alpha }_j$ and ${\beta }_j$ as unknown intercept and slope parameters for the regression within stratum $U_j.$ The errors ${\epsilon }_i\mathrm{,}i\in U_j,$ are assumed to be iid with mean 0 and finite variance ${\sigma }_{\epsilon }^2$ and to be independent of $(x_i\mathrm{,}{z}_i),$ and the variables $x_i$ for $i\in U_j$ are assumed to have finite variance. Also, to enable PPS sampling, we assume that ${\max }_{i\in U_j}{n}_j{p}_{ij}\text{<}1$ with probability approaching 1 for large $m,$ i.e., for large $n_j\mathrm{,}{N}_j.$

In this section, asymptotic properties of estimators ${\widehat{Y}}_{\text{reg}\mathrm{,}k}\mathrm{,}{\widehat{Y}}_{\text{dec}}$ are considered with respect to the regression model and repeated sampling. By Theorem 1, the model-assisted estimators ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ and ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ are still consistent and asymptotically normal for triples $(x_i\mathrm{,}{y}_i\mathrm{,}{z}_i)$ iid within strata, since the conditions (C1)-(C4) are satisfied under moment assumptions on $z_i\mathrm{,}1/z_i$ even if model (2.7) is incorrect. However, the estimators ${\widehat{Y}}_{\text{reg}\mathrm{,}k}$ are efficient when model (2.7) is correct.

Theorem 3 Assume model (2.7) along with (C1), with $E\left(x_i^4\right)\text{<}\infty \mathrm{,}E\left({\epsilon }_i^4\right)\text{<}\infty ,$ $E\left(z_i\right)\text{<}\infty ,$ and $E\left(\left(1+x_i^4\right)/z_i^3\right)\text{<}\infty .$ Then all conclusions in Theorem 1 and Theorem 2 still hold. In particular, when ${\beta }_1\ne {\beta }_2,$ ${\sigma }_1^2,$ the asymptotic variance of $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,1}}-Y\right)/N,$ is larger than ${\sigma }_2^2,$ the asymptotic variance of $\sqrt{n}\left({\widehat{Y}}_{\text{reg}\mathrm{,2}}-Y\right)/N.$ Furthermore,

$$\sqrt{n}\left({\widehat{Y}}_{\text{dec}}-Y\right)/N{\to }_{d}N\left(\mathrm{0,}\left(1-\pi \right){\sigma }_1^2+\pi {\sigma }_2^2\right)\mathrm{,} (2.8)$$

where $\pi$ is the limit of $P\left({\widehat{Y}}_{\text{dec}}={\widehat{Y}}_{\text{reg}\mathrm{,2}}\right).$

Note that $\pi$ in (2.8) is equal to 1 when ${\beta }_1\ne {\beta }_2$ and equal to $\tau$ when ${\beta }_1={\beta }_2.$

According to Theorem 3, under model (2.7), all three estimators defined in (1.2)-(1.4) have the same asymptotic efficiency when ${\alpha }_1={\alpha }_2$ and ${\beta }_1={\beta }_2$ (condition (2.3)). Furthermore, ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ is asymptotically worse than ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ when ${\beta }_1\ne {\beta }_2.$ Thus, why would we not always use ${\widehat{Y}}_{\text{reg}\mathrm{,2}}?$

The assertions in Theorem 3 are first-order asymptotic results. A more refined, second-order asymptotic result under the conditions in Theorem 3 and condition (2.3) when the sizes $z_i$ are all equal is that, up to a term of order $n_1^{-2}+n_2^{-2},$

$$\text{mse}\left(\frac{{\widehat{Y}}_{\text{reg}\mathrm{,1}}}{N}\right)-\frac{{\sigma }_{\epsilon }^2}{n}\le \left[\text{mse}\left(\frac{{\widehat{Y}}_{\text{reg}\mathrm{,2}}}{N}\right)-\frac{{\sigma }_{\epsilon }^2}{n}\right]\left[1-\frac{n_1{n}_2{\left({\overline{X}}_1-{\overline{X}}_2\right)}^2}{n{D}_n}\right]\mathrm{,} (2.9)$$

where mse is the mean squared error conditional on $x_i’\text{s},$ ${\overline{X}}_j=N_j^{-1}{\displaystyle {\sum }_{i\in U_j}}{x}_i,$ and

$$D_n={\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i\in U_j}}{\left(x_i-{\overline{X}}_j\right)}^2+\frac{n_1{n}_2{\left({\overline{X}}_1-{\overline{X}}_2\right)}^2}{n}\mathrm{.}$$

Result (2.9) indicates that, when weights are equal and ${\beta }_1={\beta }_2$ and ${\alpha }_1={\alpha }_2,$ the finite sample performance of ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ may be better than that of ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ for moderate $n_1$ and $n_2$ . See the simulation results in Section 4. The proof of (2.9) is a special case of a more general result in Slud (2012) and, thus, is omitted.

In applications, we do not know whether ${\beta }_1={\beta }_2.$ Hence, the decision-based estimator ${\widehat{Y}}_{\text{dec}}$ is an adaptive procedure to select a good estimator. In view of (2.8), the performance of ${\widehat{Y}}_{\text{dec}}$ is close to (slightly worse than) that of ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ when ${\beta }_1\ne {\beta }_2,$ and is close to (slightly worse than) that of ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ when ${\alpha }_1={\alpha }_2$ and ${\beta }_1={\beta }_2.$This is also supported by the simulation results in Section 4.

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