Local polynomial estimation for a small area mean under informative sampling
Section 1. Introduction

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Population totals and means are often required for small subpopulations (or areas). When the inference is based on the area specific sample data, the resulting small area parameter estimators (direct estimators) are not of adequate precision due to the small area specific sample sizes. As a result, it becomes necessary to borrow strength across areas. Indirect estimators (predictors) that borrow strength are obtained when a model is used for the population of small areas. The model provides a link to related small areas. As a consequence, a model-based small area indirect estimator uses all the observations in the national sample, as well as the observations from the small area.

Suppose that the population of interest, $U$ of size $N,$ consists of $M$ non-overlapping areas with $N_i$ units in the $i^{\text{th}}$ small area $U_i$ $\left(i=1,\dots ,M\right).$ A sample, $s,$ of $m$ areas is first selected using a specified sampling scheme with inclusion probabilities ${\pi }_i=m{p}_i$ $\left(i=1,\dots ,M\right),$ where $p_i$ denotes the selection probability of small area $i.$ Subsamples $s_i$ of specified sizes $n_i$ are independently selected from each small area $U_i$ according to a specified sampling design with selection probabilities $p_{j|i}$ $\left({\displaystyle {\sum }_{j=1}^{N_i}{p}_{j|i}=1}\right).$ The inclusion probabilities are ${\pi }_{j|i}=n_i{p}_{j|i}$ with sampling weights $w_{j|i}={\pi }_{j|i}^{-1}.$ We consider the selection probabilities $p_{j|i}$ proportional to a size measure, $c_{ij},$ related to the response variable $y_{ij}\text{:}$ that is $p_{j|i}=c_{ij}/{\displaystyle {\sum }_{k=1}^{N_i}{c}_{ik}}.$ We assume that all small areas are sampled, that is $m=M.$ The resulting overall sample size is $n={\displaystyle {\sum }_{i=1}^M{n}_i}.$

The basic population nested error regression model introduced by Battese, Harter and Fuller (1988) is given by

$$y_{ij}=x_{ij}^T\beta +v_i+e_{ij},j=1,\dots ,N_i;i=1,\dots ,M,(1.1)$$

where $y_{ij}$ is the value of the response variable for unit $j$ in small area $i,$ $x_{i\text{}j}={\left(1,x_{ij1},\dots ,x_{ijp}\right)}^T$ is the vector of covariates, $\beta ={\left({\beta }_0,{\beta }_1,\dots ,{\beta }_p\right)}^T$ is the vector of fixed effects, and $v_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_v^2\right)$ are the random small area effects independent of the unit level errors $e_{ij}\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_e^2\right).$ The estimation of small area means, ${\overline{Y}}_i=N_i^{-1}{\displaystyle {\sum }_{j=1}^{N_i}{y}_{ij}},$ is of primary interest.

If the sampling design is non-informative for the model, that is if the model (1.1) holds for the sample, then efficient model-based estimators of the small area means ${\overline{Y}}_i$ can be obtained using empirical best linear unbiased prediction (EBLUP) (see Rao and Molina, 2015, Chapter 6 for an excellent account of the procedure). In this case, both the sample and population models coincide, allowing the use of (1.1) on the sample data to estimate ${\overline{Y}}_i.$

If the selection probability $p_{j|i}$ is related to $y_{ij}$ even after conditioning on $x_{ij},$ the sampling design is informative and the model (1.1) no longer holds for the sample. Consequently, the EBLUP estimator, that is based on (1.1) for the sample, may be heavily biased. It is, therefore, necessary to develop estimators that can account for sample selection, thereby reducing estimation bias. To this end, Verret et al. (2015) augmented model (1.1) by including the variable $g\left(p_{j|i}\right),$ where $g\left(p_{j|i}\right)$ is a specified function of the probability $p_{j|i}.$ Their model for the sample is given by

$$y_{ij}=x_{ij}^T{\beta }_0+g\left(p_{j|i}\right){\delta }_0+v_{0i}+e_{0ij},j=1,\dots ,n_i;i=1,\dots ,M,(1.2)$$

where $v_{0i}\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_{0v}^2\right)$ and independent of $e_{0ij}\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_{0e}^2\right),$ and ${\beta }_0={\left({\beta }_{00},{\beta }_{01},\dots ,{\beta }_{0p}\right)}^T.$ Verret et al. (2015) checked the adequacy of (1.2) after fitting the model to sample data $\left(y_{ij},x_{ij},p_{j|i}\right),j=1,\dots ,n_i;i=1,\dots ,M,$ for different choices of $g(\cdot )$ that provide the best fit to the data. They suggested the following four possibilities for the choice of $g\left(p_{j|i}\right)\text{:}$ $p_{j|i},$ $\log \left(p_{j|i}\right),$ $w_{j|i}={\left(n_i{p}_{j|i}\right)}^{-1}$ and $n_i{w}_{j|i}=p_{j|i}^{-1}.$ Since their sample model is parametric, the EBLUP theory can be used to estimate the relevant parameters using model (1.2).

Verret et al. (2015) illustrated via a simulation that the resulting EBLUP estimator, denoted as ${\widehat{\overline{Y}}}_i^{\text{VRH}},$ obtained under (1.2), performs well under informative sampling design by reducing both bias and mean squared error as compared to the EBLUP estimator, ${\widehat{\overline{Y}}}_i^{\text{EBLUP}},$ obtained from the sample data under the non-augmented model (1.1). Their simulation study compared their approach to the one used in Pfeffermann and Sverchkov (2007). Their simulation results showed that the bias-adjusted estimator of Pfeffermann and Sverchkov (2007) performed well under informative sampling in terms of bias, but that its MSE is significantly larger than the corresponding MSE of the EBLUP estimator based on the augmented model.

In this paper, we make no assumptions concerning the form of the function $g\left(p_{j|i}\right).$ Instead, we incorporate the $p_{j|i}’\text{s}$ into the model (1.1) via an unknown smooth function $m_0\left(p_{j|i}\right).$ Our smooth function $m_0(\cdot )$ does not have a parametric form such as the one in Verret et al. (2015). We suppose that $m_0(\cdot )$ can be locally approximated by a polynomial of order $q.$ For each point $l$ in small area $U_k,$ the corresponding polynomial is obtained by the Taylor expansion of $m_0\left(p_{j|i}\right)$ in a neighbourhood of $p_{l|k}.$ For each point $\left(l,k\right)$ in the population, we replace $m_0\left(p_{j|i}\right)$ by the corresponding parametric approximation and fit the resulting model just as in parametric fitting. We refer to this method as parametric polynomial localization.

This local approximation results in an augmented model that is semi-parametric. Such models have been applied to small area estimation by Opsomer, Claeskens, Ranalli, Kauermann and Breidt (2008). These authors chose a technique based on penalized splines to estimate the non-parametric part of their models. Breidt and Opsomer (2000) and Breidt, Opsomer, Johnson and Ranalli (2007) used the local polynomial technique in survey sampling theory to construct model-assisted estimators. Their estimators were based on non-parametric models without random effects. To the best of our knowledge, the estimation of a small area mean ${\overline{Y}}_i,$ based on a local polynomial technique under semiparametric models has hardly been investigated.

The paper is structured as follows. Section 2 provides a review of two methods that result in estimators that account for sample selection: these methods were developed by Pfeffermann and Sverchkov (2007) and by Verret et al. (2015). In Section 3, we present a three-step procedure to estimate the proposed semi-parametric augmented model and the small area mean ${\overline{Y}}_i$ using a local polynomial approximation. We label the resulting estimator of the small area mean as ${\widehat{\overline{Y}}}_i^{\text{LP}}.$ The mean squared error (or MSE) of ${\widehat{\overline{Y}}}_i^{\text{LP}}$ is estimated in Section 4 by a parametric conditional bootstrap method. The conditional bootstrap method is also used to estimate the MSE of EBLUP estimators obtained under augmented model (1.2). In Section 5, we conduct a simulation study under the design-model (or $pm)$ framework to compare the bias and MSE of the new estimator ${\widehat{\overline{Y}}}_i^{\text{LP}}$ to the EBLUP estimator, as well as to the two estimators discussed in Verret et al. (2015). We also study the performance of the conditional bootstrap procedure in estimating the MSE of the proposed local polynomial and EBLUP estimators studied in Verret et al. (2015). The performance is evaluated in terms of mean relative bias and mean confidence interval level. Concluding remarks are given in Section 6.

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