Model-Assisted Estimation of Small Area Poverty Measures: An Application within the Valencia Region in Spain

Abstract

This paper introduces small area estimators of poverty indexes, with special attention to the poverty rate (or Head Count Index), and studies the sampling design consistency and the asymptotic normality of these estimators. The estimators are assisted by nested error regression models and are model-assisted counterparts of model-based empirical best predictors. Simulation studies show that these estimators present a good balance between sampling bias and mean squared error. Data from the 2013 Spanish living conditions survey with respect to the region of Valencia are used to determine the performance of this new method for estimating the poverty rate.

Appendix: Assumptions of Theorem 4.1

A 1

$\lim _{N\rightarrow \infty }\varvec{\theta }_N=\varvec{\theta }+O(N^{-1/2})$ and $\lim _{N\rightarrow \infty }\big (u_{dN}(\varvec{\theta }_N)-u_{dN}(\varvec{\theta })\big )=o(1)$.

A 2

$\lim _{N\rightarrow \infty }(\hat{\varvec{\theta }}-\varvec{\theta }_N)=O_\pi (n_N^{-1/2})$,

A 3

The expected sample size $n^* = E_{\pi }(n) = O(N^\delta )$, with $1/2< \delta < 1$

A 4

$K_L \le N\pi _j/n^*\le K_U$ for all j, where $K_L$ and $K_U$ are positive constants.

A 5

For any vector z with finite 2 $+$ $\lambda$ population moments with arbitrarily small $\lambda > 0$, let $\bar{z}_{HT}= \frac{1}{N} \sum _{j\in s} z_j/\pi _j$ we assume that $V_{\pi }(\bar{z}_{HT}) \le g_1 n^*(N - 1)^{-1} \sum _{j \in U} (z_j -\bar{z}_{N})(z_j -\bar{z}_{N})^\prime$ for some constant $g_1$

A 6

For any z with finite fourth population moment the Horvitz-Thompson estimators satisfy a central limit theorem:

$$\begin{aligned} (V_{\pi }(\bar{z}_{HT}))^{-1/2} (\bar{z}_{HT}-\bar{z}_{N}) \mathop {\rightarrow }\limits ^{\mathcal {L}} N(0, I_{p \times p}) \end{aligned}$$

and the estimated covariance matrix for the Horvitz-Thompson estimators is design consistent in the following sense:

$$\begin{aligned} (V_{\pi }(\bar{z}_{HT}))^{-1} \hat{V}_{HT}(\bar{z}_{HT}) - I_{p \times p} = O_{\pi }(n^{*-1/2}) \end{aligned}$$

where the design variance-covariance matrix of $\bar{z}_{HT}$ denoted by $V_{\pi }(\bar{z}_{HT})^{-1/2}$, is positive definite, and $\hat{V}_{HT}(\bar{z}_{HT})$ is the Horvitz-Thompson estimator of $V_{\pi }(\bar{z}_{HT}))^{-1/2}$.

A 7

The population level function $T_N(\varvec{\eta })= \frac{1}{N_d}\sum _{j\in U_{d}} g_{dj}(\varvec{\eta })$ converges to a limiting smooth function $T(\varvec{\eta })$, uniformly in a neighborhood of $\varvec{\theta }$. This limiting function is uniformly continuous for $\varvec{\eta }$ in a neighborhood of $\varvec{\theta }$ and has finite first and second derivatives with respect to $\varvec{\eta }$ .

A 8

The population quantity

$$\begin{aligned} sup_{\varvec{\eta }\in C} N^{\alpha } |T_N(\varvec{\theta }_N +N^{-\alpha } \varvec{\eta }) - T_N(\varvec{\theta }_N) - T(\varvec{\theta }_N +N^{-\alpha } \varvec{\eta }) +T(\varvec{\theta }_N)| \rightarrow 0 \end{aligned}$$

where C is a large enough compact set in $R^{p+2}$ and $\alpha \in (\frac{1}{4}, \frac{1}{2}]$.

These assumptions are similar to those used in Wang and Opsomer (2011) and Fabrizi et al. (2014). Assumptions A1 and A2 ensure that the sample fit $\hat{\varvec{\theta }}$ and the population fit $\varvec{\theta }_N$ share a common limit. Assumptions A3, A4, A5 and A6 are satisfied for commonly used sample size designs in reasonably finite populations. However, it would not hold for systematic sampling or one-per-stratum designs. A7 assumption about the estimator allows us to use the limiting smooth function instead of nonsmooth population quantity in asymptotic expansion.