The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables

In sample surveys, it is usual to make use of auxiliary information to increase the precision of the estimators. We propose a new chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables and obtain the mean squared error (MSE) equations for the proposed estimators. We find theoretical conditions that make proposed estimators more efficient than the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables.     

Improved Generalized Ratio-Cum-Product Estimation

In this paper, a generalized ratio-cum-product estimator for estimating the ratio (product) of two population means using auxiliary information on two other variables is given of which the estimators by SINGH (1969) and SHAH and SHAH (1978) are particular cases. The estimator is regeneralized when the covariance between two auxiliary variables is known.     

Estimating Ratio of Proportions Using Auxiliary Information

The problem of estimation of ratio of population proportions is considered and a difference-type estimator is proposed using auxiliary information. The bias and mean squared error of the proposed estimator is found and compared to the usual estimator and also to WYNN'S (1976) type estimator. An example is included for illustration.     

A Note on the Estimation of the Population Mean in Stratified Random Sampling Using Two Auxiliary Variables

In this paper, a two‐phase sampling estimator for a stratified population mean using two auxiliary variables x and z is considered when the stratum mean of x is unknown but that of z is known. The suggested estimator under its optimal condition is found to be more efficient than the one using only x.     

Small area estimation when auxiliary information is measured with error

Small area estimation methods typically combine direct estimatesfrom a survey with predictions from a model in order to obtainestimates of population quantities with reduced mean squarederror. When the auxiliary information used in the model is measuredwith error, using a small area estimator such as the Fay–Herriotestimator while ignoring measurement error may be worse thansimply using the direct estimator. We propose a new small areaestimator that accounts for sampling variability in the auxiliaryinformation, and derive its properties, in particular showingthat it is approximately unbiased. The estimator is appliedto predict quantities measured in the U.S. National Health andNutrition Examination Survey, with auxiliary information fromthe U.S. National Health Interview Survey.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

Two auxiliary variates in ratio method of estimation

A ratio type estimator using two auxiliary variates has been proposed and conditions are obtained to choose between proposed estimator and OLKIN'S (1958) estimator using two auxiliary variates.     

Estimating mean cost using auxiliary covariates

We study the estimation of mean medical cost when censoring is dependent and a large amount of auxiliary information is present. Under missing at random assumption, we propose semiparametric working models to obtain low-dimensional summarized scores. An estimator for the mean total cost can be derived nonparametrically conditional on the summarized scores. We show that when either the two working models for cost-survival process or the model for censoring distribution is correct, the estimator is consistent and asymptotically normal. Small-sample performance of the proposed method is evaluated via simulation studies. Finally, our approach is applied to analyze a real data set in health economics.     

Semiparametric inference for a 2-stage outcome-auxiliary-dependent sampling design with continuous outcome

Two-stage design has long been recognized to be a cost-effective way for conducting biomedical studies. In many trials, auxiliary covariate information may also be available, and it is of interest to exploit these auxiliary data to improve the efficiency of inferences. In this paper, we propose a 2-stage design with continuous outcome where the second-stage data is sampled with an "outcome-auxiliary-dependent sampling" (OADS) scheme. We propose an estimator which is the maximizer for an estimated likelihood function. We show that the proposed estimator is consistent and asymptotically normally distributed. The simulation study indicates that greater study efficiency gains can be achieved under the proposed 2-stage OADS design by utilizing the auxiliary covariate information when compared with other alternative sampling schemes. We illustrate the proposed method by analyzing a data set from an environmental epidemiologic study.     

A Product-Type Estimator in Double Sampling

The use of ratio and product estimators, using auxiliary information, for estimating the mean of a finite population is well known. The efficiency of ratio estimator or product estimator is high depending on whether the auxiliary character is highly positively or negatively coorelated with the main character of interest. This paper proposes a product-type estimator which is more efficient than the usual ratio and product estimators in practical situations. We consider the case of double sampling from which the single sampling results may easily be derived.     

Estimated Pseudopartial‐Likelihood Method for Correlated Failure Time Data with Auxiliary Covariates

Summary As biological studies become more expensive to conduct, statistical methods that take advantage of existing auxiliary information about an expensive exposure variable are desirable in practice. Such methods should improve the study efficiency and increase the statistical power for a given number of assays. In this article, we consider an inference procedure for multivariate failure time with auxiliary covariate information. We propose an estimated pseudopartial likelihood estimator under the marginal hazard model framework and develop the asymptotic properties for the proposed estimator. We conduct simulation studies to evaluate the performance of the proposed method in practical situations and demonstrate the proposed method with a data set from the studies of left ventricular dysfunction ( SOLVD Investigators, 1991 , New England Journal of Medicine 325 , 293–302).     

An Improved Class of Estimators for Finite Population Mean in Sample Surveys

This paper proposes a class of estimators for estimating the finite population mean -Y of a study variate y using information on two auxiliary variates, one of which is positively and the other negatively correlated with the study variate y. An “asymptotically optimum estimator” (AOE) in the class is identified with its bias and mean square error formulae. It is observed that the proposed AOE is more efficient than Srivastava (1965), Srivastava (1974), Prasad (1989) and Gandge , Varghese , and Prabhu-Ajgaonkar (1993) estimators.     

Buckley-James Estimator of AFT Models with Auxiliary Covariates

In this paper we study the Buckley-James estimator of accelerated failure time models with auxiliary covariates. Instead of postulating distributional assumptions on the auxiliary covariates, we use a local polynomial approximation method to accommodate them into the Buckley-James estimating equations. The regression parameters are obtained iteratively by minimizing a consecutive distance of the estimates. Asymptotic properties of the proposed estimator are investigated. Simulation studies show that the efficiency gain of using auxiliary information is remarkable when compared to just using the validation sample. The method is applied to the PBC data from the Mayo Clinic trial in primary biliary cirrhosis as an illustration.     

Ratio estimation with measurement error in the auxiliary variate

Summary .  With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate.     

On the inadmissibility of Watterson's estimator

We consider the estimation of the scaled mutation parameter θ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.     

Use of correlated scrambling variables in quantitative randomized response technique

In this paper, we develop a new methodology that indicates that the use of correlated scrambling variables in the randomized response technique may play an important role in increasing the efficiency of an estimator of the population mean of a sensitive variable. Although it is clear analytically that the proposed estimator is more efficient than its existing competitors, we have investigated the magnitude of the gain in efficiency through simulation studies that involve both real secondary data from the health sciences, as well as artificial data. We also derive an estimator of the variance of the proposed estimator of mean and we study the coverage of 95% confidence intervals based on this variance estimator. An application using real primary data on smoking by university students is also included.     

Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences canlead to very different estimators. These two paradigms are notso different if we search for an optimal strategy rather thanjust an optimal estimator, a strategy being a pair composedof a sampling design and an estimator. We show that, under alinear model, the optimal model-assisted strategy consists ofa balanced sampling design with inclusion probabilities thatare proportional to the standard deviations of the errors ofthe model and the Horvitz–Thompson estimator. If the heteroscedasticityof the model is 'fully explainable’ by the auxiliary variables,then this strategy is also optimal in a model-based sense. Moreover,under balanced sampling and with inclusion probabilities thatare proportional to the standard deviation of the model, thebest linear unbiased estimator and the Horvitz–Thompsonestimator are equal. Finally, it is possible to construct asingle estimator for both the design and model variance. Theinference can thus be valid under the sampling design and underthe model.     

Empirical Bayes small area prediction under a zero-inflated lognormal model with correlated random area effects

Many variables of interest in agricultural or economical surveys have skewed distributions and can equal zero. Our data are measures of sheet and rill erosion called Revised Universal Soil Loss Equation - 2 (RUSLE2). Small area estimates of mean RUSLE2 erosion are of interest. We use a zero-inflated lognormal mixed effects model for small area estimation. The model combines a unit-level lognormal model for the positive RUSLE2 responses with a unit-level logistic mixed effects model for the binary indicator that the response is nonzero. In the Conservation Effects Assessment Project (CEAP) data, counties with a higher probability of nonzero responses also tend to have a higher mean among the positive RUSLE2 values. We capture this property of the data through an assumption that the pair of random effects for a county are correlated. We develop empirical Bayes (EB) small area predictors and a bootstrap estimator of the mean squared error (MSE). In simulations, the proposed predictor is superior to simpler alternatives. We then apply the method to construct EB predictors of mean RUSLE2 erosion for South Dakota counties. To obtain auxiliary variables for the population of cropland in South Dakota, we integrate a satellite-derived land cover map with a geographic database of soil properties. We provide an R Shiny application called viscover (available at https://lyux.shinyapps.io/viscover/ ) to visualize the overlay operations required to construct the covariates. On the basis of bootstrap estimates of the mean square error, we conclude that the EB predictors of mean RUSLE2 erosion are superior to direct estimators.     

Random effects logistic regression analysis with auxiliary covariates

We study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable. We extend the semiparametric estimator of Pepe and Fleming (1991, Journal of the American Statistical Association 86, 108-113) to the random effects model using the best linear unbiased prediction approach of Henderson (1975, Biometrics 31, 423-448). The method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications. Simulation study results show that the proposed method outperforms the existing methods. We analyzed a data set from the Collaborative Perinatal Project using the proposed method and found that the use of DDT increases the risk of preterm births among U.S. children.     

Class of Estimators of Mean and Variance Using Auxiliary Information when Correlation Coefficient is also Known

For estimating the mean of a finite population using information on an auxiliary variable, a class of estimators which also uses the value of the correlation coefficient between the two variables which is assumed known, is defined. Expression for its asymptotic mean squared error and its minimum value is obtained. An expression by which the minimum mean squared error of this class is smaller than those which use only the sample mean and the sample variance of the auxiliary variable is obtained. A similar class of estimators is considered for the estimation of the population variance. The gain in efficiency is illustrated for six populations considered in literature.