Small area quantile estimation via spline regression and empirical likelihood

Section 3. Proposed approach

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For any $\alpha \in \left(\mathrm{0,}1\right),$ the ${\alpha }^{\text{th}}$ quantile of a distribution $F$ is defined to be

$${\xi }_{\alpha }\mathrm{<mo>=</mo>}\text{inf}\left\{u:F\left(u\right)\ge \alpha \right\}\mathrm{.}$$

If $\widehat{F}\left(u\right)$ is an estimate of $F\left(u\right),$ its $\alpha$-quantile is naturally estimated by

$${\widehat{\xi }}_{\alpha }=\inf \left\{u:\widehat{F}\left(u\right)\ge \alpha \right\}\mathrm{.}(3.1)$$

Under the distributional assumption on ${ϵ}_{ij},$ we have

$$\begin{array}{ll}P\left(y_{ij}\le u\right)\hfill & =\mathbb{E}\left\{P\left({\epsilon }_{ij}\le u-m_0\left(x_{ij}\right)-v_i|x_{ij}\mathrm{,}{v}_i\right)\right\}\hfill \\ \hfill & \mathrm{<mtext>\hspace{0.17em}</mtext>\; <mo>=</mo>}\mathbb{E}\left\{G_i\left(u-m_0\left(x_{ij}\right)-v_i\right)\right\}.\hfill \end{array}$$

Hence, the population distribution of the $i^{\text{th}}$ small area is given by

$$F_i\left(u\right)\mathrm{<mo>=</mo>}{N}_i^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{G}_i\left(u-m_0\left(x_{ij}\right)-v_i\right)\mathrm{.}$$

Once $G_i$ and $m_0(\cdot )$ are suitably estimated, so will be the small area quantiles.

We follow the empirical likelihood idea of Chen and Liu (2018) for estimating $G_i(\cdot ).$ Suppose the values of ${\epsilon }_{ij}$ in the sample are known. Consider a candidate $G_0$ of the form

$$G_0\left(u\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{,}j}{p}_{ij}}I\left({\epsilon }_{ij}\le u\right)\mathrm{,}$$

where $I(\cdot )$ is an indicator function and ${\displaystyle {\sum }_{i\mathrm{,}j}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 0}}^m}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{n_i}}.$ We hence have $p_{ij}\mathrm{<mo>=</mo>}d{G}_0\left({\epsilon }_{ij}\right)$ and under DRM $d{G}_i\left({\epsilon }_{st}\right)=p_{st}\exp \left\{{\theta }_i^{\prime }q\left({\epsilon }_{st}\right)\right\}$ for $i\mathrm{<mo>=</mo>\; 0,}\mathrm{1,}\dots \mathrm{,}m$ which implies

$$G_i\left(u\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{s\mathrm{,}t}{p}_{st}}\exp \left\{{\theta }_i^{\prime }q\left({\epsilon }_{st}\right)\right\}I\left({\epsilon }_{st}\le u\right)\mathrm{.}(3.2)$$

By Owen (2001), we obtain the empirical likelihood function

$$L_n\left(G_0\mathrm{,}{G}_1\mathrm{,}\dots \mathrm{,}{G}_m\right)\mathrm{<mo>=</mo>}{\displaystyle \prod _{i\mathrm{,}j}d{G}_i\left({\epsilon }_{ij}\right)}\mathrm{<mo>=</mo>}\left\{{\displaystyle \prod _{i\mathrm{,}j}{p}_{ij}}\right\}\exp \left[{\displaystyle \sum _{i\mathrm{,}j}}\left\{{\theta }_i^{\prime }q\left({\epsilon }_{ij}\right)\right\}\right]\mathrm{,}$$

where the parameter $\theta$ and $p_{ij}’\text{s}$ satisfy $p_{ij}\ge 0,$ and for $s\mathrm{<mo>=</mo>\; 0,}\mathrm{1,}\dots \mathrm{,}m,$

$${\displaystyle \sum _{i\mathrm{,}j}{p}_{ij}}\exp \left\{{\theta }_s^{\prime }q\left({\epsilon }_{ij}\right)\right\}\mathrm{<mo>=</mo>\; 1.}(3.3)$$

Note that we have used the convention ${\theta }_0\mathrm{<mo>=</mo>\; 0}$ for simpler presentation. Because $G_1\mathrm{,}\dots \mathrm{,}{G}_m$ are fully determined by ${\theta }^{\prime }\mathrm{<mo>=</mo>}\left({\theta }_1^{\prime }\mathrm{,}\dots \mathrm{,}{\theta }_m^{\prime }\right)$ and $G_0,$ we write the empirical log-likelihood as

$$l_n\left(\theta \mathrm{,}{G}_0\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{,}j}}\log \left(p_{ij}\right)+{\displaystyle \sum _{ij}{\theta }_i^{\prime }}q\left({\epsilon }_{ij}\right)\mathrm{.}$$

Maximizing $l\left(\theta \mathrm{,}{G}_0\right)$ with respect to $G_0$ under the constraints (3.3) results in fitted probabilities

$${\widehat{p}}_{ij}\mathrm{<mo>=</mo>}{n}^{-1}{\left\{1+{\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 1}}^m{\lambda }_s}\left[\exp \left\{{\theta }_s^{\prime }q\mathrm{<mo\; stretchy="false">(</mo>}{\epsilon }_{ij}\mathrm{<mo\; stretchy="false">)</mo>}\right\}-1\right]\right\}}^{-1}(3.4)$$

and the profile log EL

$$l_n\left(\theta \right)\mathrm{<mo>=</mo>}-{\displaystyle \sum _{i\mathrm{,}j}}\log \left\{1+{\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 1}}^m{\lambda }_s}\left[\exp \left\{{\theta }_s^{\prime }q\left({\epsilon }_{ij}\right)\right\}-1\right]\right\}+{\displaystyle \sum _{i\mathrm{,}j}{\theta }_i^{\prime }}q\left({\epsilon }_{ij}\right)$$

with $\left({\lambda }_1\mathrm{,}\dots \mathrm{,}{\lambda }_m\right)$ being the solution to

$${\displaystyle \sum _{s\mathrm{,}t}}\frac{\exp \left\{{\theta }_i^{\prime }q\left({\epsilon }_{st}\right)\right\}-1}{1+{\displaystyle {\sum }_{l\mathrm{<mo>=</mo>1}}^m{\lambda }_l}\left[\exp \left\{{\theta }_l^{\prime }q\left({\epsilon }_{st}\right)\right\}-1\right]}\mathrm{<mo>=</mo>0.}$$

Since the values of ${\epsilon }_{ij}$ are not available, we replace them by the residuals obtained from fitting model (2.1) under assumption (2.2):

$${\widehat{\epsilon }}_{ij}\mathrm{<mo>=</mo>}{y}_{ij}-{\widehat{m}}_0\left(x_{ij}\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\gamma }\right)-{\widehat{v}}_i$$

where

$${\widehat{m}}_0\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\gamma }\mathrm{<mo\; stretchy="false">)</mo>\; <mo>=</mo>}{\widehat{\beta }}_0+{\widehat{\beta }}_{1}x+\dots +{\widehat{\beta }}_p{x}^p+{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K{\widehat{\gamma }}_k}{\left(x-{\kappa }_k\right)}_{+}^p\mathrm{.}(3.5)$$

Let ${\widehat{l}}_n\left(\theta \right)$ be the log EL function ${\tilde{l}}_n\left(\theta \right)$ after ${\epsilon }_{ij}$ are replaced by ${\widehat{\epsilon }}_{ij}.$ We define the maximum EL estimator of $\theta$ by $\widehat{\theta }\mathrm{<mo>=</mo>}\text{argmax}{\widehat{l}}_n\left(\theta \right)$ and estimate $G_i\left(u\right)$ by

$${\tilde{G}}_i\left(u\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{s\mathrm{,}t}{\widehat{p}}_{st}}\exp \left\{{\widehat{\theta }}_i^{\prime }q\left({\widehat{\epsilon }}_{st}\right)\right\}I\left({\widehat{\epsilon }}_{st}\le u\right)(3.6)$$

with

$${\widehat{p}}_{st}\mathrm{<mo>=</mo>}{n}^{-1}{\left\{1+{\displaystyle \sum _{l\mathrm{<mo>=</mo>\; 1}}^m}\left(n_l/n\right)\left[\exp \left\{{\theta }_l^{\prime }q\left({\widehat{\epsilon }}_{st}\right)\right\}-1\right]\right\}}^{-1}$$

and ${\widehat{\theta }}_0\mathrm{<mo>=</mo>\; 0.}$ The R package $drmdel$ can be used to compute $\widehat{\theta }$ and ${\widehat{p}}_{ij}$ which has 11 choices of basis function $q\left(u\right).$

Because ${\tilde{G}}_i\left(u\right)$ is discrete, the following kernel smoothed distribution ${\widehat{G}}_i\left(u\right)$ leads to better quantile estimation:

$${\widehat{G}}_i\left(u\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_i}{\widehat{w}}_{ij}}\Phi \left(\frac{{\widehat{\epsilon }}_{ij}-u}{b}\right)\mathrm{,}(3.7)$$

where the weights are chosen to be ${\widehat{w}}_{ij}\mathrm{<mo>=</mo>}{\tilde{G}}_i\left({\widehat{\epsilon }}_{ij}\right)-{\tilde{G}}_i\left({\widehat{\epsilon }}_{ij}-\right),b$ is a bandwidth parameter, and $\Phi (\cdot )$ is the distribution function of standard normal. As suggested by Chen and Liu (2013), we choose $b\mathrm{<mo>=</mo>}\text{1}\text{.06}{n}^{-\mathrm{1\; <mo>/</mo>\; 5}}\min \left\{\widehat{\sigma }\mathrm{,}\widehat{Q}/\text{1}\text{.34}\right\}$ where $\widehat{\sigma }$ is the standard deviation of the distribution ${\widehat{G}}_i$ and $\widehat{Q}$ is its interquartile range.

In some applications, only population power means of covariates are known and can be used for statistical inference. In other applications, covariates of all members of the population are known. This leads two possible quantile estimates. In the first case, we estimate $F_i$ by

$${\widehat{F}}_i^{\left(a\right)}\left(u\right)\mathrm{<mo>=</mo>}{n}_i^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_i}{\widehat{G}}_i}\left(u-{\widehat{\overline{Y}}}_i-\left\{{\widehat{m}}_0\left(x_{ij}\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\gamma }\right)-{\widehat{m}}_0\left({\overline{x}}_i\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\gamma }\right)\right\}\right)\mathrm{,}(3.8)$$

where we use ${\widehat{m}}_0\left({\overline{x}}_i\mathrm{<mo>;</mo>}\widehat{\beta }\mathrm{,}\widehat{\gamma }\right)$ specified in (3.5).

When the census information about $x$ is available, we estimate $F_i$ by

$${\widehat{F}}_i^{\left(b\right)}\left(u\right)\mathrm{<mo>=</mo>}{N}_i^{-1}\left\{{\displaystyle \sum _{j\in s_i}I\left(y_{ij}\le u\right)}+{\displaystyle \sum _{j\in r_i}{\widehat{G}}_i\left(u-{\widehat{m}}_0\left(x_{ij}\right)-{\widehat{v}}_i\right)}\right\}\mathrm{,}(3.9)$$

where $s_i$ and $r_i$ are sets of observed and unobserved units in small area $i.$ The rest of the specifications are the same as in (3.8).

The proposed estimates resemble those of Chen and Liu (2018) but we use a non-parametric regression. Because collecting population power means of covariates is easier than collecting covariates values of all units in the population ${\widehat{F}}_i^{\left(a\right)}\left(u\right)$ is more broadly applicable than ${\widehat{F}}_i^{\left(b\right)}\left(u\right).$ It is also computationally more efficient. Because ${\widehat{F}}_i^{\left(b\right)}\left(u\right)$ uses covariate values of all units in the population, it should statistically outperform when both are applicable.

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