Nicola Salvati

Small area estimation: the EBLUP estimator based on spatially correlated random area effects

This paper deals with small area indirect estimators under area level random effect models when only area level data are available and the random effects are correlated. The performance of the Spatial Empirical Best Linear Unbiased Predictor (SEBLUP) is explored with a Monte Carlo simulation study on lattice data and it is applied to the results of the sample survey on Life Conditions in Tuscany (Italy). The mean squared error (MSE) problem is discussed illustrating the MSE estimator in comparison with the MSE of the empirical sampling distribution of SEBLUP estimator. A clear tendency in our empirical findings is that the introduction of spatially correlated random area effects reduce both the variance and the bias of the EBLUP estimator. Despite some residual bias, the coverage rate of our confidence intervals comes close to a nominal 95%.

Small area estimation for spatial correlation in watershed erosion assessment

This article describes the combination of small area estimator and a simultaneously autoregressive (SAR) model applied to the erosion data collected at the Rathbun Lake Watershed in Iowa (USA). The proposed methodology considers and EBLUP estimator with spatially correlated random area effects taking into account the information provided by neighboring areas. The article discusses the gain obtained from modeling the spatial correlation among small area random effects useful in representing the unexplained variation of the small area target quantities. Moreover the estimator of mean squared error of the proposed estimator is presented.

M-quantile models with application to poverty mapping

Over the last decade there has been growing demand for estimates of population characteristics at small area level. Unfortunately, cost constraints in the design of sample surveys lead to small sample sizes within these areas and as a result direct estimation, using only the survey data, is inappropriate since it yields estimates with unacceptable levels of precision. Small area models are designed to tackle the small sample size problem. The most popular class of models for small area estimation is random effects models that include random area effects to account for between area variations. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. An alternative approach to small area estimation that is based on the use of M-quantile models was recently proposed by Chambers and Tzavidis (Biometrika 93(2):255–268, 2006) and Tzavidis and Chambers (Robust prediction of small area means and distributions. Working paper, 2007). Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumption and automatically provide outlier robust inference. In this paper we illustrate for the first time how M-quantile models can be practically employed for deriving small area estimates of poverty and inequality. The methodology we propose improves the traditional poverty mapping methods in the following ways: (a) it enables the estimation of the distribution function of the study variable within the small area of interest both under an M-quantile and a random effects model, (b) it provides analytical, instead of empirical, estimation of the mean squared error of the M-quantile small area mean estimates and (c) it employs a robust to outliers estimation method. The methodology is applied to data from the 2002 Living Standards Measurement Survey (LSMS) in Albania for estimating (a) district level estimates of the incidence of poverty in Albania, (b) district level inequality measures and (c) the distribution function of household per-capita consumption expenditure in each district. Small area estimates of poverty and inequality show that the poorest Albanian districts are in the mountainous regions (north and north east) with the wealthiest districts, which are also linked with high levels of inequality, in the coastal (south west) and southern part of country. We discuss the practical advantages of our methodology and note the consistency of our results with results from previous studies. We further demonstrate the usefulness of the M-quantile estimation framework through design-based simulations based on two realistic survey data sets containing small area information and show that the M-quantile approach may be preferable when the aim is to estimate the small area distribution function.

Small area estimation using a nonparametric model-based direct estimator

Nonparametric regression is widely used as a method of characterizing a non-linear relationship between a variable of interest and a set of covariates. Practical application of nonparametric regression methods in the field of small area estimation is fairly recent, and has so far focussed on the use of empirical best linear unbiased prediction under a model that combines a penalized spline (p-spline) fit and random area effects. The concept of model-based direct estimation is used to develop an alternative nonparametric approach to estimation of a small area mean. The suggested estimator is a weighted average of the sample values from the area, with weights derived from a linear regression model with random area effects extended to incorporate a smooth, nonparametrically specified trend. Estimation of the mean squared error of the proposed small area estimator is also discussed. Monte Carlo simulations based on both simulated and real datasets show that the proposed model-based direct estimator and its associated mean squared error estimator perform well. They are worth considering in small area estimation applications where the underlying population regression relationships are non-linear or have a complicated functional form.

Nonparametric M-quantile regression using penalised splines

Quantile regression investigates the conditional quantile functions of a response variable in terms of a set of covariates. M-quantile regression extends this idea by a ‘quantile-like’ generalisation of regression based on influence functions. In this work, we extend it to nonparametric regression, in the sense that the M-quantile regression functions do not have to be assumed to have a certain parametric form, but can be left undefined and estimated from the data. Penalised splines are employed to estimate them. This choice makes it easy to move to bivariate smoothing and semiparametric modelling. An algorithm based on iteratively reweighted penalised least squares to actually fit the model is proposed. Quantile crossing is addressed using an a posteriori adjustment to the function fits following He [1]. Simulation studies show the finite sample properties of the proposed estimation technique.

Small Area Model-Based Estimators Using Big Data Sources

Abstract The timely, accurate monitoring of social indicators, such as poverty or inequality, on a finegrained spatial and temporal scale is a crucial tool for understanding social phenomena and policymaking, but poses a great challenge to official statistics. This article argues that an interdisciplinary approach, combining the body of statistical research in small area estimation with the body of research in social data mining based on Big Data, can provide novel means to tackle this problem successfully. Big Data derived from the digital crumbs that humans leave behind in their daily activities are in fact providing ever more accurate proxies of social life. Social data mining from these data, coupled with advanced model-based techniques for fine-grained estimates, have the potential to provide a novel microscope through which to view and understand social complexity. This article suggests three ways to use Big Data together with small area estimation techniques, and shows how Big Data has the potential to mirror aspects of well-being and other socioeconomic phenomena.

Sleep disturbances in elderly subjects: an epidemiological survey in an Italian district

Bonanni E, Tognoni G, Maestri M, Salvati N, Fabbrini M, Borghetti D, Di Coscio E, Choub A, Sposito R, Pagni C, Iudice A, Murri L. Sleep disturbances in elderly subjects: an epidemiological survey in an Italian district.
Acta Neurol Scand: 2010: 122: 389–397.
© 2010 The Authors Journal compilation

Small area estimation under spatial nonstationarity

A geographical weighted empirical best linear unbiased predictor (GWEBLUP) for a small area average is proposed, and an estimator of its conditional mean squared error is developed. The popular empirical best linear unbiased predictor under the linear mixed model is obtained as a special case of the GWEBLUP. Empirical results using both model-based and design-based simulations, with the latter based on two real data sets, show that the GWEBLUP predictor can lead to efficiency gains when spatial nonstationarity is present in the data. A practical gain from using the GWEBLUP is in small area estimation for out of sample areas. In this case the efficient use of geographical information can potentially improve upon conventional synthetic estimation.

Robust small area prediction for counts

A new semiparametric approach to model-based small area prediction for counts is proposed and used for estimating the average number of visits to physicians for Health Districts in Central Italy. The proposed small area predictor can be viewed as an outlier robust alternative to the more commonly used empirical plug-in predictor that is based on a Poisson generalized linear mixed model with Gaussian random effects. Results from the real data application and from a simulation experiment confirm that the proposed small area predictor has good robustness properties and in some cases can be more efficient than alternative small area approaches.

Small area estimation of the mean using non-parametric M-quantile regression: a comparison when a linear mixed model does not hold

The demand for reliable statistics in subpopulations, when only reduced sample sizes are available, has promoted the development of small area estimation methods. In particular, an approach that is now widely used is based on the seminal work by Battese et al. [An error-components model for prediction of county crop areas using survey and satellite data, J. Am. Statist. Assoc. 83 (1988), pp. 28–36] that uses linear mixed models (MM). We investigate alternatives when a linear MM does not hold because, on one side, linearity may not be assumed and/or, on the other, normality of the random effects may not be assumed. In particular, Opsomer et al. [Nonparametric small area estimation using penalized spline regression, J. R. Statist. Soc. Ser. B 70 (2008), pp. 265–283] propose an estimator that extends the linear MM approach to the case in which a linear relationship may not be assumed using penalized splines regression. From a very different perspective, Chambers and Tzavidis [M-quantile models for small area estimation, Biometrika 93 (2006), pp. 255–268] have recently proposed an approach for small-area estimation that is based on M-quantile (MQ) regression. This allows for models robust to outliers and to distributional assumptions on the errors and the area effects. However, when the functional form of the relationship between the qth MQ and the covariates is not linear, it can lead to biased estimates of the small area parameters. Pratesi et al. [Semiparametric M-quantile regression for estimating the proportion of acidic lakes in 8-digit HUCs of the Northeastern US, Environmetrics 19(7) (2008), pp. 687–701] apply an extended version of this approach for the estimation of the small area distribution function using a non-parametric specification of the conditional MQ of the response variable given the covariates [M. Pratesi, M.G. Ranalli, and N. Salvati, Nonparametric m-quantile regression using penalized splines, J. Nonparametric Stat. 21 (2009), pp. 287–304]. We will derive the small area estimator of the mean under this model, together with its mean-squared error estimator and compare its performance to the other estimators via simulations on both real and simulated data.