Small area quantile estimation via spline regression and empirical likelihood
Section 1. Introduction
Sample surveys are widely used to obtain information about totals, means, medians and other quantities of finite populations. Likewise, similar information on sub-populations such as individuals in specific areas and socio-demographic groups are also of interest. Often, a survey is designed to collect information of interest at the population level but leads to insufficient direct information on sub-populations. Because of this, estimating sub-population parameters with satisfactory precision and evaluating their accuracy pose serious challenges to statisticians. Statisticians must resort to suitable models to pool the information across small areas in order to properly estimate parameters for small areas when only small samples or no samples in these areas are available from the sample survey.
Research on small area estimation has received increased attention from both public and private sectors. As historical remarks, we refer to Fay and Herriot (1979), Battese, Harter and Fuller (1988), Prasad and Rao (1990), and Lahiri and Rao (1995) among many others. For a general review of the developments in small area estimation, we refer to Pfeffermann (2002) and Pfeffermann (2013) and the books of Rao (2003) and Rao and Molina (2015). See also Jiang and Lahiri (2006a), Jiang and Lahiri (2006b) and Jiang (2010) for recent publications.
Compared to quantiles, there are relatively more research activities on estimating small area means. Studies on small area quantile estimation are gaining ground. The M-quantile approach of Chambers and Tzavidis (2006) has achieved substantial success. This approach uses the M-quantile approach to characterize the conditional distributions of the response variable given covariates This information is then used to predict unobserved response values based on which the small area population distributions are estimated. Small area quantile estimation is a natural and welcome side-benefit. See Tzavidis and Chambers (2005), Pratesi, Ranalli and Salvati (2008), Tzavidis, Salvati and Pratesi (2008), and Salvati, Tzavidis and Pratesi (2012) for these developments.
Another approach for small area quantile estimation is proposed by Molina (2010). Let and be the sets of sampled and non-sampled units in a survey and and be vectors of corresponding response values. Under a parametric assumption on the joint distribution of and (or the transformed responses) they proposed to work out the conditional distribution of given (and other information). After having the joint distribution and therefore the conditional distribution properly estimated, they suggested sampling from the estimated conditional distribution to create an artificial but complete population with unobserved filled up. The population distribution is estimated based on the completed population. This approach works well for estimating small area means and quantiles. Other methods we are aware of include Tzavidis, Marchetti and Chambers (2010), Chaudhuri and Ghosh (2011) and Chen and Liu (2018). Tzavidis et al. (2010) proposed a general framework for robust small area estimation, based on representing a small area estimator as a function of a predictor of this small area cumulative distribution function. Chaudhuri and Ghosh (2011) proposed an empirical likelihood based Bayesian method. Chen and Liu (2018) proposed an approach for populations admitting a nested-error linear regression model combined with error distributions satisfying a semi-parametric density ratio model (DRM). Simulations indicate that the DRM-based method stands out when the error distributions are skewed.
In this paper, we are interested in the situation where the regression function is not linear, although the nested-error regression model remains appropriate similar to Opsomer et al. (2008). Clearly, methods derived under linear models may lead to substantial bias if the linearity assumption is violated. To reduce the potential risk of serious bias, Opsomer et al. (2008) proposed an Empirical Best Linear Unbiased Prediction (EBLUP) for the small area means under a non-parametric regression model via penalized splines (P-splines); Jiang, Ngueyen and Rao (2010) developed an adaptive fence approach employing a non-parametric model selection technique; Sperlich and José Lombardía (2010) used the local polynomial inference method in the context of small area estimation; Rao, Sinha and Dumitrescu (2014) proposed a robust EBLUP under a P-splines approximated mixed model; Torabi and Shokoohi (2015) proposed a unified analysis of both discrete and continuous responses under P-spline regression models.
We follow their lead and extend their results to allow non-normal error distributions in the nested-error non-parametric regression model. More specifically, we assume the nested-error non-parametric regression model but relax the small area error distribution assumption from normal to a flexible semi-parametric DRM. We use the P-splines regression approach of Opsomer et al. (2008) to fit the nonlinear regression. Empirical likelihood is then applied to estimate the parameters in the DRM based on the residuals. This leads to natural area specific error distribution estimation. A kernel method is then applied to obtain smoothed estimates of error distributions and small area quantiles. We construct quantile estimates in two situations: one is where we have knowledge of only covariate power means at the population level, the other is where we have covariate values of all sample units in the population. Our approach should inherit the merits of working under a non-parametric regression model, and gain from avoiding a parametric error distribution assumption. The resulting small area quantile estimates are hence more robust. Simulations indicate that when the regression function is approximately linear, the performance of the proposed approach is competitive. The proposed approach outperforms when the regression relationship is quadratic or exponential.
The rest of the paper is organized as follows. Section 2 introduces the model and assumptions. Section 3 presents the proposed approach. Section 4 proposes a bootstrap procedure for estimating mean squared errors. In Section 5, we use Monte Carlo methods to evaluate the performance of the proposed method and compare it with some existing methods. An application example is reported in Section 6. Section 7 contains some concluding remarks.
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