Search

We present the first far-UV (FUV) imaging results of the intermediate-age Galactic open cluster (OC) NGC 2818 that has a Planetary nebula (PN) within the field. We explore whether the PN is a member of the cluster using images taken from the Ultra-violet Imaging Telescope (UVIT) aboard AstroSat. We identify the cluster members by combining the UVIT-detected sources with Gaia EDR3 data. We detect four bright and hot blue straggler stars (BSSs) and two yellow straggler stars (YSSs) based on their location in the optical and FUV-optical color-magnitude diagrams (CMDs). The theoretical isochrones more or less fit the observed distribution of detected stars in all the CMDs. Based on the parameters estimated using Spectral Energy Distribution (SED), we infer that the BSSs are either collisional products or might have undetectable white dwarf (WD) companions. Our photometric analysis of YSSs confirms their binarity, consistent with the spectroscopic results. We find the YSSs to be formed through a mass-transfer scenario and the hot components are likely to be A-type subdwarfs. A comparison of the radial velocity (RV), Gaia EDR3 proper-motion (PM) of the PN with the cluster members, and reddening towards the PN and the cluster does not rule out the membership of the PN. Using SED, the estimated stellar parameters of the PN’s central star match well with the previous estimations. Comparing the central star’s position with theoretical pAGB models suggest that it has already entered the WD cooling phase, and its mass is deduced to be ∼ 0.6Msun. The corresponding progenitor mass turns out to be ∼ 2.1Msun, comparable to the turn-off mass of the cluster, implying that the progenitor could have formed in the cluster. We suggest that the NGC 2818 might be one of the few known clusters to host a PN, providing a unique opportunity to test stellar evolution models.

Two-level models are widely used for analysing clustered survey data with the design structure matching the model hierarchy. Hypothesis testing on model parameters is studied, using a weighted composite likelihood approach that takes account of the survey design features. In particular, the asymptotic normality of the weighted composite likelihood estimators is established. Using this result, the asymptotic distributions of a generalised score test statistic and a likelihood ratio type test statistic, under a null composite hypothesis on the model parameters, is established. Results of a limited simulation study on the finite sample performance of the proposed tests are reported.

Survey samplers have long been using probability samples from one or more sources in conjunction with census and administrative data to make valid and efficient inferences on finite population parameters. This topic has received a lot of attention more recently in the context of data from non-probability samples such as transaction data, web surveys and social media data. In this paper, I will provide a brief overview of probability sampling methods first and then discuss some recent methods, based on models for the non-probability samples, which could lead to useful inferences from a non-probability sample by itself or when combined with a probability sample. I will also explain how big data may be used as predictors in small area estimation, a topic of current interest because of the growing demand for reliable local area statistics.

Summary We propose a Bayesian empirical likelihood approach to survey data analysis on a vector of finite population parameters defined through estimating equations. Our method allows overidentified estimating equation systems and is applicable to both smooth and non-differentiable estimating functions. Our proposed Bayesian estimator is design consistent for general sampling designs and the Bayesian credible intervals are calibrated in the sense of having asymptotically valid design-based frequentist properties under single-stage unequal probability sampling designs with small sampling fractions. Large sample properties of the Bayesian inference proposed are established for both non-informative and informative priors under the design-based framework. We also propose a Bayesian model selection procedure with complex survey data and show that it works for general sampling designs. An efficient Markov chain Monte Carlo procedure is described for the required computation of the posterior distribution for general vector parameters. Simulation studies and an application to a real survey data set are included to examine the finite sample performances of the methods proposed as well as the effect of different types of prior and different types of sampling design.

In this short note, I describe my memorable interactions with Professor C. R. Rao starting from my student days at the University of Bombay (1954–1956) to my stay at the Indian Statistical Institute, 1968–1969, as a visiting professor.

ENTHIS LINK GOES TO A ENGLISH SECTIONFRTHIS LINK GOES TO A FRENCH SECTION Missing observations due to non‐response are commonly encountered in data collected from sample surveys. The focus of this article is on item non‐response which is often handled by filling in (or imputing) missing values using the observed responses (donors). Random imputation (single or fractional) is used within homogeneous imputation classes that are formed on the basis of categorical auxiliary variables observed on all the sampled units. A uniform response rate within classes is assumed, but that rate is allowed to vary across classes. We construct confidence intervals (CIs) for a population parameter that is defined as the solution to a smooth estimating equation with data collected using stratified simple random sampling. The imputation classes are assumed to be formed across strata. Fractional imputation with a fixed number of random draws is used to obtain an imputed estimating function. An empirical likelihood inference method under the fractional imputation is proposed and its asymptotic properties are derived. Two asymptotically correct bootstrap methods are developed for constructing the desired CIs. In a simulation study, the proposed bootstrap methods are shown to outperform traditional bootstrap methods and some non‐bootstrap competitors under various simulation settings. The Canadian Journal of Statistics 47: 281–301; 2019 © 2019 Statistical Society of Canada Supporting Information

Area-level unmatched sampling and linking models have been widely used as a model-based method for producing reliable estimates of small-area means. However, one practical difficulty is the specification of a link function. In this paper, we relax the assumption of a known link function by not specifying its form and estimating it from the data. A penalized-spline method is adopted for estimating the link function, and a hierarchical Bayes method of estimating area means is developed using a Markov chain Monte Carlo method for posterior computations. Results of simulation studies comparing the proposed method with a conventional approach based on a known link function are presented. In addition, the proposed method is applied to data from the Survey of Family Income and Expenditure in Japan and poverty rates in Spanish provinces.

In this short article, I will attempt to provide some highlights of my chancy life as a statistician in chronological order spanning over 60 years, 1954 to present.

The widely used Elbers–Lanjouw–Lanjouw (ELL) method of estimating complex parameters for areas with small sample sizes uses a fitted nested‐error model based on survey data to create simulated censuses of the variable of interest. The complex parameters obtained from each simulated censuses are then averaged to get the estimate. An empirical best (EB) method, under the nested‐error model with normal errors, is significantly more efficient, in terms of mean square error (MSE), than the ELL method when the normality assumption holds. However, it can perform poorly in terms of MSE when the model errors are not normally distributed. We relax normality by assuming skew‐normal errors, derive EB estimators, and study their MSE relative to EB based on normality and ELL. We propose bootstrap methods for MSE estimation. We also study an improvement to ELL by conditioning on the area random effects and without parametric assumptions on the errors.

An efficient and convenient procedure has been described for one-pot multi-component synthesis of tetrahydrobenzo[b]pyrans known as 2-amino-7,7-dimethyl-5-oxo-4-phenyl-5,6,7,8-tetrahydro-4H-chromene-3-carbonitrile which can be obtained from the reaction of substituted aromatic aldehydes, dimedone, malonitrile, in the presence of base such as potassium tertiary butoxide and THF in methanol as solvent at RT condition. All the compounds were examined by advanced spectroscopic data ( 1 H NMR, 13 C NMR and LCMS) and the structural determination was evaluated by elemental analysis. In addition to this, all the newly synthesized compounds were examined for their antibacterial activities and antifungal activity by disc diffusion method against the organism of Aspergillus niger and Candida ablicans L. KEY WORDS : Aromatic aldehydes, Dimedone, Malonitrile, Potasium tertiary butoxide, 2-Amino-7,7-dimethyl-5-oxo-4-phenyl-5,6,7,8-tetrahydro-4H-chromene-3-carbonitrile, Anti-microbial activity Bull. Chem. Soc. Ethiop. 2018 , 32(1), 133-138 DOI: https://dx.doi.org/10.4314/bcse.v32i1.12

The authors use an empirical Bayes (EB) approach to small area estimation under area-level unmatched sampling and linking models. Model parameters are estimated by a unified expectation and maximization (EM) algorithm and used to obtain EB estimators of area parameters. Results are extended to a nonparametric linking model based on a spline approximation. Approximate EB estimators that are computationally simpler are also obtained. Different bootstrap approaches to estimating the mean squared error (MSE) of the EB estimators are proposed. Results of a simulation study on the performance of the proposed methods are presented. Proposed methods are applied to data from a survey of family income and expenditure in Japan and poverty rates in Spanish provinces.

Summary Poverty maps at local level might be misleading when based on direct (or area-specific) estimators obtained from a survey that does not cover adequately all the local areas of interest. In this case, small area estimation procedures based on assuming common models for all the areas typically provide much more reliable poverty estimates. These models include area effects to account for the unexplained between-area heterogeneity. When poverty figures are sought at two different aggregation levels, domains and subdomains, it is reasonable to assume a twofold nested error model including random effects explaining the heterogeneity at the two levels of aggregation. The paper introduces the empirical best (EB) method for poverty mapping or, more generally, for estimation of additive parameters in small areas, under a twofold model. Under this model, analytical expressions for the EB estimators of poverty incidences and gaps in domains or subdomains are given. For more complex additive parameters, a Monte Carlo algorithm is used to approximate the EB estimators. The EB estimates obtained of the totals for all the subdomains in a given domain add up to the EB estimate of the domain total. We develop a bootstrap estimator of the mean-squared error of EB estimators and study the effect on the mean-squared error of a misspecification of the area effects. In simulations, we compare the estimators obtained under the twofold model with those obtained under models with only domain effects or only subdomain effects, when all subdomains are sampled or when there are unsampled subdomains. The methodology is applied to poverty mapping in counties of the Spanish region of Valencia by gender. Results show great variation in the poverty incidence and gap across the counties from this region, with more counties affected by extreme poverty when restricting ourselves to women.

We present a simple yet effective variable selection method for the two-fold nested subarea model, which generalizes the widely-used Fay-Herriot area model. The twofold subarea model consists of a sampling model and a linking model, which has a nested-error model structure but with unobserved responses. To select variables under the two-fold subarea model, we first transform the linking model into a model with the structure of a regular regression model and unobserved responses. We then estimate an information criterion based on the transformed linking model and use the estimated information criterion for variable selection. The proposed method is motivated by the variable selection method of Lahiri and Suntornchost (2015) for the Fay-Herriot model and the variable selection method of Li and Lahiri (2019) for the unit-level nested-error regression model. Simulation results show that the proposed variable selection method performs significantly better than some naive competitors, especially when the variance of the area-level random effect in the linking model is large.

Public-use survey data are an important source of information for researchers in social science and health studies to build statistical models and make inferences on the target finite population. This paper presents two general inferential tools through the pseudo empirical likelihood and the sample empirical likelihood methods. Theoretical results on point estimation and linear or nonlinear hypothesis tests involving parameters defined through estimating equations are established, and practical issues with the implementation of the proposed methods are discussed. Results from simulation studies and an application to the 2016 General Social Survey dataset of Statistics Canada show that the proposed methods work well under different scenarios. The inferential procedures and theoretical results presented in the paper make the empirical likelihood a practically useful tool for users of complex survey data., Comment: 50 pages, including 11 pages of tables

Standard statistical methods that do not take proper account of the complexity of survey design can lead to erroneous inferences when applied to survey data due to unequal selection probabilities, clustering, and other design features. In particular, the actual type I error rates of tests of hypotheses using standard methods can be much bigger than the nominal significance level. Methods that take account of survey design features in testing hypotheses have been proposed, including Wald tests and quasi-score tests that involve the estimated covariance matrices of parameter estimates. In this paper, we present a unified approach to hypothesis testing that does not require computing the covariance matrices by constructing bootstrap approximations to weighted likelihood ratio statistics and weighted quasi-score statistics and establish the asymptotic validity of the proposed bootstrap tests. In addition, we also consider hypothesis testing from categorical data and present a bootstrap procedure for testing simple goodness of fit and independence in a two-way table. In the simulation studies, the type I error rates of the proposed approach are much closer to their nominal significance level compared with the naive likelihood-ratio test and quasi-score test. An application to data from an educational survey under a logistic regression model is also presented.

In this paper, we, for the first time, report geochemistry of sandstone from Somanpalli Group from Pomburna area in the Eastern Belt of Pranhita–Godavari (PG) Valley, central India and studied to infer their provenance, intensity of paleo-weathering and depositional tectonic setting. Petrographic study of sandstones show QFL modal composition of arenite. Chemical results show high $$\hbox {SiO}_{2}$$ and CIA but lower $$\hbox {Al}_{2}\hbox {O}_{3}, \hbox {TiO}_{2}$$ , Rb, Sr, $$\hbox {K}_{2}\hbox {O}$$ indicating mixed sources. Major elements chemistry parameters such as, $$\hbox {K}_{2}\hbox {O/Al}_{2}\hbox {O}_{3}$$ ratio and positive correlation of Rb with $$\hbox {K}_{2}\hbox {O}$$ , reflects a warm and humid climate for study area. The tectonic discrimination plots ( $$\hbox {SiO}_{2}/20$$ – $$\hbox {K}_{2}\hbox {O} + \hbox {Na}_{2}\hbox {O}$$ – $$\hbox {TiO}_{2} + \hbox {Fe}_{2}\hbox {O}_{3} + \hbox {MgO};\,\hbox {K}_{2}\hbox {O}/\hbox {Na}_{2}\hbox {O}$$ vs. $$\hbox {SiO}_{2}$$ ; Th–Sc–Zr/20) indicate dominantly passive margin and slight active tectonic setting. Concentrations of Zr, Nb, Y, and Th are higher compared to the UCC values and the trends in Th/Cr, Th/Co, La/Sc and Cr/Zr ratios support a felsic and mafic source for these sandstones and deposition in passive margin basin. Chondrite normalized REE pattern reflects LREE depletion, negative Eu anomaly and flat HREE similar to UCC, felsic components. ICV value (0.95) also supports tectonically quiescent passive margin settings. CIA values (74) indicate high degree of chemical weathering and warm and humid paleoclimatic condition.

Small area estimation has been extensively studied under unit level linear mixed models. In particular, empirical best linear unbiased predictors (EBLUPs) of small area means and associated estimators of mean squared prediction error (MSPE) that are unbiased to second order have been developed. However, EBLUP can be sensitive to outliers. Sinha & Rao (2009) developed a robust EBLUP method and demonstrated its advantages over the EBLUP in the presence of outliers in the random small area effects and/or unit level errors in the model. A bootstrap method for estimating MSPE of the robust EBLUP was also proposed. In this paper, we relax the assumption of linear regression for the fixed part of the model and we replace it by a weaker assumption of a semi-parametric regression. By approximating the semi-parametric mixed model by a penalized spline mixed model, we develop robust EBLUPs of small area means and bootstrap estimators of MSPE. Results of a simulation study are also presented. The Canadian Journal of Statistics 42: 126–141; 2014 © 2013 Statistical Society of Canada Resume L'estimation pour petits domaines a ete largement etudiee a l'aide de modeles lineaires mixtes au niveau des unites. Dans ce contexte, les meilleurs predicteurs lineaires sans biais empiriques (MPLSBE) pour la moyenne de petits domaines ont ete developpes, ainsi que des estimateurs sans biais au deuxieme ordre pour l'erreur quadratique moyenne de prevision (EQMP) associee. Cependant, les MPLSBE peuvent etre sensibles aux donnees aberrantes. Sinha et Rao (2009) ont elabore des MPLSBE robustes et ont demontre leurs avantages par rapport aux MPLSBE en presence de donnees aberrantes dans les effets aleatoires des petits domaines ou dans les termes d'erreur au niveau de l'unite. Ces auteurs ont aussi propose une methode bootstrap d'estimation de l'EQMP des MPLSBE robustes. Dans cet article, les auteurs assouplissent l'hypothese de regression lineaire dans la partie fixe du modele et la remplacent par une hypothese moins rigide de regression semi-parametrique. En approximant le modele mixte semi-parametrique par un modele mixte de splines penalise, ils developpent des MPLSBE robustes pour petits domaines et des estimateurs bootstrap de l'EQMP. Les resultats d'une etude de simulation sont egalement presentes. La revue canadienne de statistique 42: 126–141; 2014 © 2013 Societe statistique du Canada

Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) introduced several bootstrap methods under adaptive cluster sampling using a Horvitz–Thompson type estimator. Using a simulation study, they showed that their proposed methods provide confidence intervals with highly understated coverage rates. In this article, we first show that their bootstrap methods provide biased bootstrap estimates. We then define two bootstrap methods, based on the method of Gross (Proceeding of the survey research methods section. American Statistical Association, Alexandria, VA, pp 181–184, 1980) and Bootstrap With Replacement, that provide unbiased bootstrap estimates of the population mean with bootstrap variances matching the corresponding unbiased variance estimator. Using a simulation study, we show that the bootstrap confidence intervals based on our proposed methods have better performance than those based on available bootstrap methods, in the sense of having coverage proportion closer to the nominal coverage level. We also compare the proposed intervals to empirical likelihood based intervals in small samples.

Combining information from two or more independent surveys is a problem frequently encountered in survey sampling. We consider the case of two independent surveys, where a large sample from survey 1 collects only auxiliary information and a much smaller sample from survey 2 provides information on both the variables of interest and the auxiliary variables. We propose a model-assisted projection method of estimation based on a working model, but the reference distribution is design-based. We generate synthetic or proxy values of a variable of interest by first fitting the working model, relating the variable of interest to the auxiliary variables, to the data from survey 2 and then predicting the variable of interest associated with the auxiliary variables observed in survey 1. The projection estimator of a total is simply obtained from the survey 1 weights and associated synthetic values. We identify the conditions for the projection estimator to be asymptotically unbiased. Domain estimation using the projection method is also considered. Replication variance estimators are obtained by augmenting the synthetic data file for survey 1 with additional synthetic columns associated with the columns of replicate weights. Results from a simulation study are presented. Copyright 2012, Oxford University Press.

Charged particles and photons have been measured in central Pb + Pb collisions at 158 A GeV in a common (eta-phi)-phase space region in the WA98 experiment at the CERN SPS. The measured distributions have been analyzed to quantify the frequency with which phase space regions of varying sizes have either small or large neutral pion fraction. The measured results are compared with VENUS model simulated events and with mixed events. Events with both large and small charged-neutral fluctuations are observed to occur more frequently than expected statistically, as deduced from mixed events, or as predicted by model simulations, with the difference becoming more prominent with decreasing size of the A Delta eta-Delta phi region. (C) 2011 Elsevier B.V. All rights reserved.

Variance estimation in the presence of imputed data has been widely studied in the literature. It is well known that treating the imputed values as if they were true values could lead to serious un...

Gini coefficient is among the most popular and widely used measures of income inequality in economic studies, with various extensions and applications in finance and other related areas. This paper studies confidence intervals on the Gini coefficient for simple random samples, using normal approximation, bootstrap percentile, bootstrap-t and the empirical likelihood method. Through both theory and simulation studies it is shown that the intervals based on normal or bootstrap approximation are less satisfactory for samples of small or moderate size than the bootstrap-calibrated empirical likelihood ratio confidence intervals which perform well for all sample sizes. Results for stratified random sampling are also presented.

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.

Using survey weights, You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] proposed a pseudo-empirical best linear unbiased prediction (pseudo-EBLUP) estimator of a small area mean under a nested error linear regression model. This estimator borrows strength across areas through a linking model, and makes use of survey weights to ensure design consistency and preserve benchmarking property in the sense that the estimators add up to a reliable direct estimator of the mean of a large area covering the small areas. In this article, a second-order approximation to the mean squared error (MSE) of the pseudo-EBLUP estimator of a small area mean is derived. Using this approximation, an estimator of MSE that is nearly unbiased is derived; the MSE estimator of You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] ignored cross-product terms in the MSE and hence it is biased. Empirical results on the performance of the proposed MSE estimator are also presented. The Canadian Journal of Statistics 38: 598–608; 2010 © 2010 Statistical Society of Canada

Pseudo empirical likelihood ratio confidence intervals for finite population parameters are based on asymptotic χ 2 approximation to an adjusted pseudo empirical likelihood ratio statistic, with the adjustment factor related to the design effect. The calculation of the design effect involves variance estimation and hence requires second order inclusion probabilities. It also depends on how auxiliary information is used, and needs to be derived one-at-a-time for different scenarios. This paper presents bootstrap procedures for constructing pseudo empirical likelihood ratio confidence intervals. The proposed method bypasses the need for design effects and is valid under general single-stage unequal probability sampling designs with small sampling fractions. Different scenarios in using auxiliary information are handled by simply including the same type of benchmark constraints with the bootstrap procedures. Simulation results show that the bootstrap calibrated intervals perform very well and have much improved coverage probabilities over the χ 2 -based intervals when the sample sizes are small or moderate.

Summary Bayesian methods for inference on finite population means and other parameters by using sample survey data face hurdles in all three phases of the inferential procedure: the formulation of a likelihood function, the choice of a prior distribution and the validity of posterior inferences under the design-based frequentist framework. In the case of independent and identically distributed observations, the profile empirical likelihood function of the mean and a non-informative prior on the mean can be used as the basis for inference on the mean and the resulting Bayesian empirical likelihood intervals are also asymptotically valid under the frequentist set-up. For complex survey data, we show that a pseudo-empirical-likelihood approach can be used to construct Bayesian pseudo-empirical-likelihood intervals that are asymptotically valid under the design-based set-up. The approach proposed compares favourably with a full Bayesian analysis under simple random sampling without replacement. It is also valid under general single-stage unequal probability sampling designs, unlike a full Bayesian analysis. Moreover, the approach is very flexible in using auxiliary population information and can accommodate two scenarios which are practically important: incorporation of known auxiliary population information for the construction of intervals by using the basic design weights; calculation of intervals by using calibration weights based on known auxiliary population means or totals.

Estimation of small area means under a basic area level model is studied, using an empirical Bayes (best) estimator or a weighted estimator with fixed weights. Mean squared errors (MSEs) of the estimators and nearly unbiased (or exactly unbiased) estimators of MSE are derived under three different approaches: design based (approach 1), unconditional model based (approach 2) and conditional model based (approach 3). Performance of MSE estimators under the three approaches with respect to relative bias and coefficient of variation is also studied, using a simulation experiment.

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model-based and design-based simulation studies. Results show large reductions in mean squared error relative to direct area-specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada

This article presents a pseudo–empirical likelihood approach to inference for multiple-frame surveys. We establish a unified framework for point and interval estimation of finite population parameters, and show that inferences on the parameters of interest making effective use of different types of auxiliary population information can be conveniently carried out through the constrained maximization of the pseudo–empirical likelihood function. Confidence intervals are constructed using either the asymptotic χ2 distribution of an adjusted pseudo–empirical likelihood ratio statistic or a bootstrap calibration method. Simulation results based on Statistics Canada’s Family Expenditure Survey data show that the proposed methods perform well in finite samples for both point and interval estimation. In particular, a multiplicity-based pseudo–empirical likelihood method is proposed. This method is easily used for multiple-frame surveys with more than two frames and does not require complete frame membership inform...

Variance estimation after imputation is an important practical problem in survey sampling. When deterministic imputation or stochastic imputation is used, we show that the variance of the imputed estimator can be consistently estimated by a unifying linearize and reverse approach. We provide some applications of the approach to regression imputation, fractional categorical imputation, multiple imputation and composite imputation. Results from a simulation study, under a factorial structure for the sampling, response and imputation mechanisms, show that the proposed linearization variance estimator performs well in terms of relative bias, assuming a missing at random response mechanism. Copyright 2009, Oxford University Press.

Small area estimation has received considerable attention in recent years because of growing demand for small area statistics. Basic area-level and unit-level models have been studied in the literature to obtain empirical best linear unbiased prediction (EBLUP) estimators of small area means. Although this classical method is useful for estimating the small area means efficiently under normality assumptions, it can be highly influenced by the presence of outliers in the data. In this article, the authors investigate the robustness properties of the classical estimators and propose a resistant method for small area estimation, which is useful for downweighting any influential observations in the data when estimating the model parameters. To estimate the mean squared errors of the robust estimators of small area means, a parametric bootstrap method is adopted here, which is applicable to models with block diagonal covariance structures. Simulations are carried out to study the behaviour of the proposed robust estimators in the presence of outliers, and these estimators are also compared to the EBLUP estimators. Performance of the bootstrap mean squared error estimator is also investigated in the simulation study. The proposed robust method is also applied to some real data to estimate crop areas for counties in Iowa, using farm-interview data on crop areas and LANDSAT satellite data as auxiliary information. The Canadian Journal of Statistics 37: 381–399; 2009 © 2009 Statistical Society of Canada L'estimation de petits domaines a rec cu considerablement d'attention ces dernieres annees en raison de la demande croissante de statistiques regionales. Les modeles au niveau des domaines et des unites ont deja ete etudies dans la litterature et les meilleurs estimateurs lineaires sans biais empiriques (EBLUP) pour les petits domaines ont ete obtenus. Quoique cette methode classique est utile pour estimer les moyennes regionales de fac con efficace sous l'hypothese de normalite, ses resultats sont grandement influences par la presente de donnees aberrantes. Dans cet article, les auteurs etudient les proprietes de robustesse des estimateurs classiques et ils proposent une methode robuste pour l'estimation de petits domaines qui diminue le poids associe aux observations influentes lors de l'estimation des parametres du modele. Afin d'estimer l'erreur quadratique moyenne des estimateurs robustes des moyennes regionales, une methode d'auto-amorc cage parametrique est utilisee. Cette methode peut etre utilisee aux modeles dont la structure de covariance est bloc diagonale. Des simulations sont faites pour etudier le comportement des estimateurs robustes proposes en presence de valeurs aberrantes et aussi pour les comparer aux estimateurs EBLUP. La performance de l'estimateur “boostrap” de l'erreur quadratique moyenne est aussi etudiee dans cette etude de simulations. Cette methode robuste est appliquee a l'estimation de la superficie des cultures pour les comtes de l'Iowa en se basant sur des entrevues au niveau des fermes et en utilisant les donnees provenant du satellite LANDSAT comme information auxiliaire. La revue canadienne de statistique 37: 381–399; 2009 © 2009 Societe statistique du Canada

Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.

Summary Two-phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s1 is drawn according to a specific sampling design p(s1), and auxiliary data x are observed for the units i∈s1. Given the first-phase sample s1, a second-phase sample s2 is selected from s1 according to a specified sampling design {p(s2∣s1) }, and (y, x) is observed for the units i∈s2. In some cases, the population totals of some components of x may also be known. Two-phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson-type variance estimators are used for variance estimation. However, the Horvitz–Thompson (Horvitz & Thompson, J. Amer. Statist. Assoc. 1952) variance estimator in uni-phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non-negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy (Sen, J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy, J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two-phase sampling, assuming fixed first-phase sample size and fixed second-phase sample size given the first-phase sample. We apply the new variance estimators to two-phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy-type variance estimators of the two-phase regression estimators that make use of the first-phase auxiliary data and known population totals of some of the auxiliary variables.

Empirical Bayes predictors of small area parameters of interest are often obtained under a linear mixed model for continuous response data or a generalized linear mixed model for binary responses or count data. However, estimation of the unconditional mean squared error of prediction is complicated, particularly for a nonlinear mixed model. Jiang et al. (2002) proposed a jackknife method for estimating the unconditional mean squared error and showed that the resulting estimator is nearly unbiased. The leading term of this estimator does not depend on the area-specific responses in the nonlinear case, whereas the posterior variance of the small area parameter given the model parameters is area-specific. Rao (2003) proposed an alternative method that leads to a computationally simpler jackknife estimator with an area-specific leading term. We show that a modification of Rao's method leads to a nearly unbiased area-specific jackknife estimator, which is also nearly unbiased for the conditional mean squared error given the area-specific responses. We examine the relative performances of the jackknife estimators, conditionally as well as unconditionally, in a simulation study, and apply the proposed method to estimate small area mean squared errors in disease mapping problems. Copyright 2009, Oxford University Press.