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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.

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

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

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

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.

The quality of survey estimates is directly affected by survey errors that include sampling errors due to selecting a sample rather than the whole population, and non-sampling errors arising from data collection and processing procedures. The latter include frame error, measurement error and non-response. This paper addresses design issues related to total survey error and its components. Methods for handling frame problems and non-response are also presented.

The authors show how an adjusted pseudo-empirical likelihood ratio statistic that is asymptoti- cally distributed as a chi-square random variable can be used to construct confidence intervals for a finite population mean or a finite population distribution function from complex surv ey samples. They consider both non-stratified and stratified sampling designs, with or without auxiliary in formation. They examine the behaviour of estimates of the mean and the distribution function at specifi c points using simulations calling on the Rao-Sampford method of unequal probability sampling without replacement. They conclude that the pseudo-empirical likelihood ratio confidence intervals are super ior to those based on the normal approximation, whether in terms of coverage probability, tail error rates or average length of the intervals.

The authors study the estimation of domain totals and means under survey-weighted regression imputation for missing items. They use two different approaches to inference: (i) design-based with uni form response within classes; (ii) model-assisted with ignorable response and an imputation model. They show that the imputed domain estimators are biased under (i) but approximately unbiased under (ii). They obtain a bias-adjusted estimator that is approximately unbiased under (i) or (ii). They also derive lineariza tion variance estimators. They report the results of a simulation study on the bias ratio and efficiency of alternative estimators, including a complete case estimator that requires the knowledge of response indica tors.

If a population contains many zero values and the sample size is not very large, the traditional normal approximation-based confidence intervals for the population mean may have poor coverage probabilities. This problem is substantially reduced by constructing parametric likelihood ratio intervals when an appropriate mixture model can be found. In the context of survey sampling, however, there is a general preference for making minimal assumptions about the population under study. The authors have therefore investigated the coverage properties of nonparametric empirical likelihood confidence intervals for the population mean. They show that under a variety of hypothetical populations, these intervals often outperformed parametric likelihood intervals by having more balanced coverage rates and larger lower bounds. The authors illustrate their methodology using data from the Canadian Labour Force Survey for the year 2000. Lorsque l'on ne dispose que d'un petit echantillon d'une population caracterisee par la presence de nombreuses valeurs nulles, l'intervalle de confiance pour la moyenne decoulant du theoreme central limite ne jouit pas toujours du bon taux de couverture. Ce probleme peut etre enraye en bonne partie en ayant recours a des intervalles deduits du rapport de vraisemblances parametriques associe a un modele de melange adequat. Dans le contexte d'une enquete, cependant, on prefere generalement reduire au minimum les hypotheses formulees a propos de la population sous etude. C'est ce qui a amene les auteurs a etudier les proprietes de couverture d'intervalles de confiance non parametriques pour la moyenne construits a partir de la vraisemblance empirique, us montrent que pour diverses populations hypothetiques, ces intervalles sont souvent preferables aux intervalles deduits de vraisemblances parametriques, car leur borne inferieure est plus grande et leur taux de couverture mieux equilibre. Les auteurs illustrent leur methodologie a l'aide de donnees issues de l'enquete canadienne sur la population active en l'an 2000.

The authors develop a small area estimation method using a nested error linear regression model and survey weights. In particular, they propose a pseudo-empirical best linear unbiased prediction (pseudo-EBLUP) estimator to estimate small area means. This estimator borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. The proposed estimator also has a nice self-benchmarking property. The authors also obtain an approximation to the model mean squared error (MSE) of the proposed estimator and a nearly unbiased estimator of MSE. Finally, they compare the proposed estimator with the EBLUP estimator and the pseudo-EBLUP estimator proposed by Prasad & Rao (1999), using data analyzed earlier by Battese, Harter & Fuller (1988). Emploi de poids d'echantillonnage pour I'estimation des moyennes de petits domaines au moyen du meilleur predicteur lineaire sans biais pseudo-empirique Les auteurs developpent une methode d' estimation pour petits domaines a partir de poids d' echantillonnage et d' un modele de regression lineaire a erreurs emboǐtees. Ils proposent plus speci-fiquement l' emploi du meilleur predicteur lineaire sans biais pseudo-empirique (pseudo-MPLSB) pour l' estimation des moyennes de petits domaines. Parce qu' il s' appuie sur un modele, l' estimateur en question gagne en precision sur l' ensemble des domaines. Comme il tient compte des poids d' echantillonnage, il est asymptotiquement convergent par rapport au plan d' echantillonnage. Lorsqu' agrege a la population entiere, il coincide en outre avec l' estimateur obtenu par la regression classique. Les auteurs montrent aussi comment approximer l' erreur quadratique moyenne associee au modele et l' estimer presque sans biais. Its comparent enfin le nouvel estimateur au MPLSB et au pseudo-MPLSB de Prasad & Rao (1999) au moyen de donnees deja analysees par Battese, Harter & Fuller (1988).

The missing response problem in linear regression is studied. An adjusted empirical likelihood approach to inference on the mean of the response variable is developed. A non-parametric version of Wilks's theorem for the adjusted empirical likelihood is proved, and the corresponding empirical likelihood confidence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator with asymptotic normality is defined and an adjusted empirical log-likelihood function with asymptotic X 2 is derived. A simulation study is conducted to compare the adjusted empirical likelihood methods and the normal approximation methods in terms of coverage accuracies and average lengths of the confidence intervals. Based on biases and standard errors, a comparison is also made between the empirical likelihood-based estimator and related estimators by simulation. Our simulation indicates that the adjusted empirical likelihood methods perform competitively and the use of auxiliary information provides improved inferences.

The authors use a hierarchical Bayes approach to area level unmatched sampling and Unking models for small area estimation. Empirically they compare inferences under unmatched models with those obtained under the customary matched sampling and linking models. They apply the proposed method to Canadian census undercoverage estimation, developing a full hierarchical Bayes approach using Markov Chain Monte Carlo sampling methods. They show that the method can provide efficient model-based estimates. They use posterior predictive distributions to assess model fit. Estimation pour petits domaines a I'aide d'echantillons et de modeles de liens non apparies Les auteurs proposent une approche bayesienne hierarchique pour l'estimation de petits domaines au moyen de modeles d'echantillonnage et de lien non apparies. Ils comparent experimentalement l'inference effectiee sous des modeles apparies et non apparies aux plans d'echantillonnage. Ils ap-pliquent leur methode a l'estimation du sous-denombrement dans le cadre du recensement canadien, probleme pour lequel ils elaborent une approche bayesienne hierarchique complete au moyen de methodes d'echantillonnage de Monte-Carlo par chaine de Markov. Ils montrent que leur technique conduit a des estimations efficaces et se servent de lois predictives a posteriori pour evaluer l'adequation du modele sous-jacent.

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log-likelihood ratio is asymptotically a weighted sum of chi-square variables with unknown weights. They obtain an adjusted empirical log-likelihood ratio which is asymptotically standard chi-square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log-likelihood ratio and use its distribution to approximate that of the empirical log-likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.

Suppose that several different imperfect instruments and one perfect instrument are used independently to measure some characteristic of a population. The authors consider the problem of combining this information to make statistical inference on parameters of interest, in particular the population mean and cumulative distribution function. They develop maximum empirical likelihood estimators and study their asymptotic properties. They also present simulation results on the finite sample efficiency of these estimators. RESUME Supposons qu'un instrument parfait et plusieurs instruments imparfaits soient utilises independamment pour mesurer une certaine caracteristique d'une population. Lesauteurs cherchent les moyens de synthetiser cette information de facon a pouvoir faire de l'inference statistique sur les parametres d'interět, dont la fonction de repartition et la moyenne de la population. Ils preisent la forme des estimateurs a vraisemblance em-pirique maximale et en etudient le comportement asymptotique. Ils en examinent aussi l'efficacite dans de petits echantillons par voie de simulation

A simple method is proposed for analysing grouped count data exhibiting overdispersion relative to a Poisson model. The method is similar to the approach suggested for the analysis of clustered binary data in Rao and Scott (1992). It requires no specific model for the overdispersion and it can be implemented easily using standard programs designed to handle independent Poisson counts, after a small amount of preprocessing.

Imputation is commonly used to compensate for missing data in surveys. We consider the general case where the responses on either the variable of interest y or the auxiliary variable x or both may be missing. We use ratio imputation for y when the associated x is observed and different imputations when x is not observed. We obtain design-consistent linearization and jackknife variance estimators under uniform response. We also report the results of a simulation study on the efficiencies of imputed estimators, and relative biases and efficiencies of associated variance estimators.

Recent developments in sample survey theory include the following topics: foundational aspects of inference, resampling methods for variance and confidence interval estimation, imputation for nonresponse and analysis of complex survey data. An overview and appraisal of some of these developments are presented. Des developpements recents dans la theorie d'etude des echantillons incluent les sujets suivants: Les aspects fondateurs de l'inference, les methodes de re-echantillonage pour la variance et l'estimation des intervalles de confiance, l'imputation pour la non-reponse et l'analyse de donnees d'etude complexes. Nous presentons ici un apercu et une evaluation de certains de ces developpements.

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are presentes. Un modele impliquant des effets aleatoires autocorreles et des erreurs d'echantillonnages est propose pour l'estimation des petites surfaces, utilisant a la fois des series chronologiques et des donnees transversales. Les erreurs d'echantillonnages sont presumees avoir une matrice connue de variance-covariance bloc diagonale. Ce modele est une extension d'un modele bien connu du a Fay et Herriot (1979) pour donnees transversales. Un estimateur a deux niveaux pour la moyenne d'une petite surface pour la periode en cours est obtenu sous les hypotheses du modele propose avec autocorrelation connue, en derivant d'abord l'estimateur de la meilleure prediction lineaire non biaisee (MPLNB), en assumant connues les variances et en les remplacant par leurs estimateurs consistants. Generalisant l'approche de Prasad et Rao (1986, 1990) pour le modele de Fay-Herriot, on a obtenu un estimateur de l'erreur quadratique moyenne (EQM) de l'estimateur a deux niveaux, qui est une bonne approximation d'ordre deux lorsque le nombre de points dans le temps, T, est petit ou moderement grand, et que le nombre de petites surfaces, m, est relativement grand. Le cas ou l'autocorrelation est inconnue, est aussi considere. Des resultats limites bases sur des etudes de simulations et portant sur l'efficacite des estimateurs a deux niveaux et la precision de l'EQM, sont presentes.

Inference on the regression parameters in a heteroscedastic linear regression model with replication is considered, using either the ordinary least-squares (OLS) or the weighted least-squares (WLS) estimator. A delete-group jackknife method is shown to produce consistent variance estimators irrespective of within-group correlations, unlike the delete-one jackknife variance estimators or those based on the customary 8-method assuming within-group independence. Finite-sample properties of the delete-group variance estimators and associated confidence intervals are also studied through simulation.

Mutations affecting qualitative traits were induced by seed ageing in lettuce. The mutant plants were isolated in the A2 generation and included chlorophyll-deficient types (chlorotica, lutescens, chlorina-virescens, luteo and viridalbo maculate), and morphological variants (dwarf and narrow, thick and curly leaf types). The leaf mutants were found to be either partially or completely sterile. Segregation pattern of the mutants in A3 generation showed that, except for the maculata types, all chlorophyll deficiencies and the dwarf mutant are controlled by single recessive nuclear genes. The genetic status of the leaf mutants was not clear, due to possible pleiotropic effect of the mutant genes in inducing gametophytic sterility. The maculata mutants exhibited sorting out of the normal and chlorophyll deficient regions during vegetative development and segregated for different degrees of chlorophyll deficiency in selfed progenies. The maculata mutants probably originated by plastome mutations induced by nuclear mutator genes.

Summary We re-examine the criteria of “hyper-admissibility” and “necessary bestness”, for the choice of estimator, from the point of view of their relevance to the design of actual surveys. Both these criteria give rise to a unique choice of estimator (viz. the Horvitz-Thompson estimator ỸHT) whatever be the character under investigation or sample design. However, we show here that the “principal hyper-surfaces” (or “domains”) of dimension one (which are practically uninteresting)play the key role in arriving at the unique choice. A variance estimator v1(ỸHT) (due to Horvitz-Thompson), which takes negative values “often”, is shown to be uniquely “hyperadmissible” in a wide class of unbiased estimators of the variance of ỸHT. Extensive empirical evidence on the superiority of the Sen-Yates-Grundy variance estimator v2(ỸHT) over v1(ỸHT) is presented.

GIVEN is a finite population of N units with characteristics Yt (t = 1,2, ..., N) whose total Y = Yi +Y2 + . .. +YN is to be estimated. If a sample of size n is to be drawn from such a population, it is often advantageous to select the units with unequal probability. For example, such a procedure may be useful when measures of sizes xt are known for all N units in the population which are positively correlated with the characteristics yt. In such cases, one may utilize the knowledge of the xt by selecting units with probabilities proportional to sizes xt, although this is, of course, not the only way of using the known xt. Of the literature on sampling with unequal probabilities and without replacement we mention papers by Horvitz and Thompson (1952), Narain (1951), Yates and Grundy (1953), Des Raj (1956) and Hartley and Rao (1962). There are some limitations, of varying importance, attached to all these methods. Briefly speaking, the method of Horvitz and Thompson (1952) is applicable only under severe restrictions on the prescribed probabilities, the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956) require a cumbersome evaluation of working probabilities, and Hartley and Rao (1962) give only asymptotic variance formulae for the estimates of Y for large and moderate size populations N. The present method is an attempt to avoid all these disadvantages at the expense of a slight loss in efficiency. It has the following properties: (i) It permits the computation of an estimator of the population total which has always a smaller variance than the standard estimator in sampling with unequal probabilities and with replacement. (ii) Unlike the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956), the present method does not entail heavy computations, even for sample size n > 2, for drawing the sample or computation of the estimator and its variance estimate.