Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 2. EBLUP and pseudo-EBLUP estimation

Consider the one-fold nested error regression model

$y_{i j} = x_{i j}^{T} β + v_{i} + e_{i j}, i = 1, \dots, m; j = 1, \dots, N_{i}, (2.1)$

where $y_{i j}$ is the variable of interest for the $j^{th}$ population unit in the $i^{th}$ small area, $x_{i j} = {(x_{i j 1}, \dots, x_{i j p})}^{T}$ is a vector of auxiliary variables with $x_{i j 1} = 1,$ $β = {(β_{1}, \dots, β_{p})}^{T}$ is a $p \times 1$ vector of regression parameters and $N_{i}$ is the number of population units in the $i^{th}$ small area, $U_{i}$ . The random small area effects $v_{i}$ are assumed to be i.i.d. $N (0, σ_{v}^{2}),$ and independent of the unit errors $e_{i j},$ which are assumed i.i.d. $N (0, σ_{e}^{2}) .$ We draw samples $s_{i}$ of size $n_{i}$ independently within each small area $i,$ according to a specified sampling design with first-order inclusion probabilities denoted by $π_{i j},$ for $j = 1, \dots, N_{i} .$ The total sample size is $n,$ where $n = \sum_{i = 1}^{m} n_{i} .$ The resulting basic design weights are given by $d_{i j} = 1 / π_{i j} .$ We assume that the sample design is ignorable, and that selection bias is absent. This implies that model (2.1) also holds for the sample data:

$y_{i j} = x_{i j}^{T} β + v_{i} + e_{i j}, i = 1, \dots, m; j = 1, \dots, n_{i}, (2.2)$

Model (2.2) is a special case of the general linear mixed model. Defining $y_{i} = {(y_{i 1}, \dots, y_{i n_{i}})}^{T},$ $X_{i} = {(x_{i 1}^{T}, \dots, x_{i n_{i}}^{T})}^{T},$ $v = {(v_{1}, \dots, v_{m})}^{T}$ and $e_{i} = {(e_{i 1}, \dots, e_{i n_{i}})}^{T},$ it follows that model (2.2) can be expressed in a matrix form by stacking the observations. The resulting equation is

$y = X β + Z v + e, (2.3)$

where $y = {col}_{1 \leq i \leq m} (y_{i}),$ $X = {col}_{1 \leq i \leq m} (X_{i}),$ $Z = {diag}_{1 \leq i \leq m} {1_{n_{i}}}$ and $e = {col}_{1 \leq i \leq m} (e_{i}),$ with $1_{n_{i}}$ a vector of dimension $n_{i}$ composed of ones. We denote by $G$ and $R$ the variance matrices of the random vectors $v$ and $e$ respectively. Then $G = σ_{v}^{2} I_{m}$ and $R = σ_{e}^{2} I_{n} .$ It follows that the variance matrix of vector $y,$ denoted as $V,$ is given by $V = R + Z G Z^{T} .$

The parameters of interest are the small area means ${\bar{Y}}_{i},$ where ${\bar{Y}}_{i} = N_{i}^{- 1} \sum_{j = 1}^{N_{i}} y_{i j}, i = 1, \dots, m .$ If $N_{i}$ is large, the sampling fraction $f_{i} = N_{i}^{- 1} n_{i},$ of the $i^{th}$ small area is negligible. This set-up corresponds to the case of an infinite population or negligible sampling rates. It follows that the small area means ${\bar{Y}}_{i}$ can be approximated by $μ_{i}$ (see Rao and Molina, 2015, page 174), where $μ_{i} = {\bar{X}}_{i}^{T} β + v_{i}$ and ${\bar{X}}_{i} = \sum_{j = 1}^{N_{i}} x_{i j} / N_{i}$ is the vector of population means of the $x_{i j} ’ s$ for the $i^{th}$ area. An estimator of $μ_{i}$ is given by ${\hat{μ}}_{i} = {\bar{X}}_{i}^{T} \hat{β} + {\hat{v}}_{i}$ (Rao and Molina, 2015, page 175), where $\hat{β}$ and ${\hat{v}}_{i}$ are estimators of $β$ and $v_{i}$ respectively. If $N_{i}$ is not large enough or if the sampling rates $f_{i}$ are not negligible, parameters ${\bar{Y}}_{i}$ cannot be approximated by linear combinations of $β$ and $v_{i} .$ This corresponds to the case of a finite population. Let $r_{i}$ be the set of the $N_{i} - n_{i}$ unobserved $y$ -values in small area $i .$ If we assume that we know the $x_{i j} ’ s$ for each individual in the population, an estimator ${\hat{\bar{Y}}}_{i}$ of ${\bar{Y}}_{i}$ is based on the observed values $y_{i j}, j \in s_{i},$ and predicted values ${\hat{y}}_{i j} = x_{i j}^{T} \hat{β} + {\hat{v}}_{i}$ for $j \in r_{i} .$ That is, estimator ${\hat{\bar{Y}}}_{i}$ is given by

${\hat{\bar{Y}}}_{i} = \frac{1}{N_{i}} (\sum_{j \in s_{i}} y_{i j} + \sum_{j \in r_{i}} {\hat{y}}_{i j}) . (2.4)$

Much of the SAE theory deals with the infinite population case, whereas the literature on the finite population case is more limited. In this paper we focus on finite population (or non‑negligible sampling rates) case, thereby constructing estimators based on (2.4).

2.1 EBLUP estimation

We denote by $\tilde{β}$ and $\tilde{v} = {({\tilde{v}}_{1}, \dots, {\tilde{v}}_{m})}^{T}$ the BLUP predictors of $β$ and $v$ respectively. These estimators are given by $\tilde{β} = {(X^{T} V^{- 1} X)}^{- 1} X^{T} V^{- 1} y$ and $\tilde{v} = G Z^{T} V^{- 1} (y - X \tilde{β}) .$ Under the normality assumption of $e$ and $v,$ it can be shown that $\tilde{β}$ and $\tilde{v}$ can be obtained by maximizing the joint density of $y$ and $v$ with respect to $β$ and $v .$ This is equivalent to minimizing the function

$ϕ = {(y - X β - Z v)}^{T} R^{- 1} (y - X β - Z v) + v^{T} G^{- 1} v . (2.5)$

This leads to the following mixed model equations

$A (\begin{array}{l} \tilde{β} \\ \tilde{v} \end{array}) = b, (2.6)$

where

$A = (\begin{array}{l} X^{T} R^{- 1} X X^{T} R^{- 1} Z \\ Z^{T} R^{- 1} X Z^{T} R^{- 1} Z + G^{- 1} \end{array}) and b = (\begin{array}{l} X^{T} R^{- 1} y \\ Z^{T} R^{- 1} y \end{array}) . (2.7)$

(see Rao and Molina, 2015, page 99 for details). The variance components $(σ_{v}^{2}, σ_{e}^{2})$ in equations (2.6) and (2.7) are generally unknown. Three methods of estimation, FC, ML and REML, are commonly used in SAE to estimate the variance components $(σ_{v}^{2}, σ_{e}^{2}) .$ A well-known difficulty with these methods is that the estimate of $σ_{v}^{2}$ can take on negative values. This estimate is truncated to zero when this occurs, that is ${\hat{σ}}_{v}^{2}$ is set to 0. Empirical versions of $A$ and $b,$ denoted as $\hat{A}$ and $\hat{b},$ are obtained if the unknown variance components $(σ_{v}^{2}, σ_{e}^{2})$ are replaced by estimators $({\hat{σ}}_{v}^{2}, {\hat{σ}}_{e}^{2}) .$ It follows from equation (2.6) that EBLUP estimators of model parameters $(β, v),$ denoted as $\hat{β}$ and $\hat{v} = {({\hat{v}}_{1}, \dots, {\hat{v}}_{m})}^{T},$ are given by

$(\begin{array}{l} \hat{β} \\ \hat{v} \end{array}) = {\hat{A}}^{- 1} \hat{b} . (2.8)$

Using (2.8), it can be proved that $\hat{β}$ and $\hat{v}$ are

$(\begin{array}{l} \hat{β} \\ \hat{v} \end{array}) = (\begin{array}{l} {(X^{T} {\hat{V}}^{- 1} X)}^{- 1} X^{T} {\hat{V}}^{- 1} y \\ \hat{G} Z^{T} {\hat{V}}^{- 1} (y - X \hat{β}) \end{array}), (2.9)$

where $\hat{G} = {\hat{σ}}_{v}^{2} I_{m}$ and $\hat{V} = {\hat{σ}}_{e}^{2} I_{n} + {\hat{σ}}_{v}^{2} Z Z^{T} .$

Remark 1. It is easier to invert matrices $\hat{G} = {\hat{σ}}_{v}^{2} I_{m}$ and $\hat{R} = {\hat{σ}}_{e}^{2} I_{n}$ than $\hat{V} .$ Consequently, it is simpler to use the mixed model equations (2.8) than equations (2.9) for computing $\hat{β}$ and $\hat{v} .$ However, when ${\hat{σ}}_{v}^{2}$ is equal to zero, equations (2.8) cannot be used because the ${\hat{G}}^{- 1}$ term in matrix $\hat{A}$ does not exist. In such cases, $\hat{β}$ and $\hat{v}$ can only be computed using (2.9).

Under model (2.2), it can be shown that $\hat{β}$ and ${\hat{v}}_{i}$ in $\hat{v} = {({\hat{v}}_{1}, \dots, {\hat{v}}_{m})}^{T}$ satisfy

$\sum_{i = 1}^{m} \sum_{j \in s_{i}} x_{i j} (y_{i j} - x_{i j}^{T} \hat{β} - {\hat{v}}_{i}) = 0. (2.10)$

Estimators $\hat{β}$ and ${\hat{v}}_{i}$ are used to compute EBLUP predictions ${\hat{y}}_{i j}^{EBLUP}$ for the $N_{i} - n_{i}$ unobserved units in small area $i$ : ${\hat{y}}_{i j}^{EBLUP} = x_{i j}^{T} \hat{β} + {\hat{v}}_{i}$ for $j \in r_{i} .$ An EBLUP estimator of ${\bar{Y}}_{i},$ denoted as ${\hat{\bar{Y}}}_{i}^{EBLUP},$ is obtained by replacing in (2.4) ${\hat{y}}_{i j}$ by ${\hat{y}}_{i j}^{EBLUP} .$ It follows that ${\hat{\bar{Y}}}_{i}^{EBLUP}$ is

${\hat{\bar{Y}}}_{i}^{EBLUP} = \frac{1}{N_{i}} [\sum_{j \in s_{i}} y_{i j} + x_{i r}^{T} \hat{β} + (N_{i} - n_{i}) {\hat{v}}_{i}], (2.11)$

where $x_{i r} = \sum_{j \in r_{i}} x_{i j}$ represents the sum of non sampled values $x_{i j} .$

2.2 You-Rao estimation

You and Rao (2002) proposed a pseudo-EBLUP small area mean estimator (YR estimator) that incorporates the design weights $d_{i j}$ into the formula of the EBLUP estimator. A property of the pseudo-EBLUP estimator is that the design consistency is preserved as the area sample size increases. Furthermore, the YR predictor offers protection against model failure or an informative sampling design (see among others Hidiroglou and Estevao, 2016 and Verret, Rao and Hidiroglou, 2015 for details). Pseudo EBLUP estimators can be constructed using the procedure in You and Rao (2002) with survey weights $w_{i j}$ that may be calibrated on some vector of auxiliary variables. Let ${\hat{β}}^{YR}$ and ${\hat{v}}^{YR} = {({\hat{v}}_{1}^{YR}, \dots, {\hat{v}}_{m}^{YR})}^{T}$ be the YR estimators of $β$ and $v$ respectively based on weights $w_{i j}$ (see You and Rao, 2002 for details). The estimators ${\hat{β}}^{YR}$ and ${\hat{v}}_{i}^{YR}$ satisfy the estimating unit-level based equations

$\sum_{i = 1}^{m} \sum_{j \in s_{i}} w_{i j} x_{i j} (y_{i j} - x_{i j}^{T} {\hat{β}}^{YR} - {\hat{v}}_{i}^{YR}) = 0. (2.12)$

Equations (2.12) represent the survey-weighted version of equations (2.10). You-Rao predictions ${\hat{y}}_{i j}^{YR}$ of $y_{i j}$ are computed as ${\hat{y}}_{i j}^{YR} = x_{i j}^{T} {\hat{β}}^{YR} + {\hat{v}}_{i}^{YR}$ for $j \in r_{i} .$ Replacing ${\hat{y}}_{i j}$ by ${\hat{y}}_{i j}^{YR}$ in (2.4) leads to the YR estimator of ${\bar{Y}}_{i}$ in the case of non negligible sampling rates:

${\hat{\bar{Y}}}_{i}^{YR} = \frac{1}{N_{i}} [\sum_{j \in s_{i}} y_{i j} + x_{i r}^{T} {\hat{β}}^{YR} + (N_{i} - n_{i}) {\hat{v}}_{i}^{YR}] . (2.13)$

Estimators ${\hat{β}}^{YR}$ and ${\hat{v}}^{YR}$ can alternatively be obtained as solutions to weighted mixed model equations similar to (2.6) (see Huang and Hidiroglou, 2003 for details). To this end, we define matrices $W_{i} = {diag}_{1 \leq j \leq n_{i}} {w_{i j}},$ $W = {diag}_{1 \leq i \leq m} {W_{i}}$ and $Ω = {diag}_{1 \leq i \leq m} {ω_{i}},$ where $ω_{i} = \sum_{j \in s_{i}} w_{i j}^{2} / \sum_{j \in s_{i}} w_{i j}$ for $i = 1, \dots, m .$ Let $ϕ_{w}$ be the sample weighted version of $ϕ,$ where

$ϕ_{w} = {(y - X β - Z v)}^{T} W^{1 / 2} R^{- 1} W^{1 / 2} (y - X β - Z v) + v^{T} Ω^{1 / 2} G^{- 1} Ω^{1 / 2} v, (2.14)$

with $W^{1 / 2}$ and $Ω^{1 / 2}$ representing the square root of matrices $W$ and $Ω$ respectively. In the first term of $ϕ_{w},$ the model error associated with the observation $y_{i j}$ is weighted by the corresponding survey weight $w_{i j},$ whereas in the second term of $ϕ_{w},$ the factor $ω_{i}$ in the diagonal matrix $Ω$ represents the weight attached to the small area effect $v_{i} .$ It can be shown that the minimization of $ϕ_{w}$ with respect to $β$ and $v$ leads to $({\hat{β}}^{YR}, {\hat{v}}^{YR}) .$ It follows that $({\hat{β}}^{YR}, {\hat{v}}^{YR})$ are given by

$(\begin{array}{l} {\hat{β}}^{YR} \\ {\hat{v}}^{YR} \end{array}) = {\hat{A}}_{w}^{- 1} {\hat{b}}_{w}, (2.15)$

where the known values of $A_{w}$ and $b_{w}$ are given by

$A_{w} = (\begin{array}{l} X^{T} W^{1 / 2} R^{- 1} W^{1 / 2} X X^{T} W^{1 / 2} R^{- 1} W^{1 / 2} Z \\ Z^{T} W^{1 / 2} R^{- 1} W^{1 / 2} X Z^{T} W^{1 / 2} R^{- 1} W^{1 / 2} Z + Ω^{1 / 2} G^{- 1} Ω^{1 / 2} \end{array}) and b_{w} = (\begin{array}{l} X^{T} W^{1 / 2} R^{- 1} W^{1 / 2} y \\ Z^{T} W^{1 / 2} R^{- 1} W^{1 / 2} y \end{array}), (2.16)$

and ${\hat{A}}_{w}$ and ${\hat{b}}_{w}$ are empirical versions of $A_{w}$ and $b_{w}$ obtained by estimating $G$ and $R$ by $\hat{G} = {\hat{σ}}_{v}^{2} I_{m}$ and $\hat{R} = {\hat{σ}}_{e}^{2} I_{n}$ respectively. Equation (2.15) can alternatively be written as

$(\begin{array}{l} {\hat{β}}^{YR} \\ {\hat{v}}^{YR} \end{array}) = (\begin{array}{l} {(X^{T} {\hat{V}}_{w}^{- 1} X)}^{- 1} X^{T} {\hat{V}}_{w}^{- 1} y \\ {\hat{G}}_{ω} Z^{T} {\hat{V}}_{w}^{- 1} (y - X {\hat{β}}^{YR}) \end{array}), (2.17)$

where ${\hat{G}}_{ω} = Ω^{- 1 / 2} \hat{G} Ω^{- 1 / 2}$ and ${\hat{V}}_{w} = W^{- 1 / 2} \hat{R} W^{- 1 / 2} + Z {\hat{G}}_{ω} Z^{T} .$

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Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 2. EBLUP and pseudo-EBLUP estimation

2.1 EBLUP estimation

2.2 You-Rao estimation

Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible Section 2. EBLUP and pseudo-EBLUP estimation

2.1 EBLUP estimation

2.2 You-Rao estimation

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 2. EBLUP and pseudo-EBLUP estimation