Unbiased estimators of finite population variance using auxiliary information in sample surveys

We present some unbiased estimators of the population variance in a finite population sample survey using the knowledge of population variance of an auxiliary character.Exact variance expressions for the proposed estimators are obtained and compared with usual unbiased estimator and the ratio estimator envisaged by Isaki (1983). Generalization of the proposed estimator is also suggested.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Estimation of Finite Population Mean in Stratified Random Sampling with Two Auxiliary Variables under Double Sampling Design

In this article, a chain ratio-product type exponential estimator is proposed for estimating finite population mean in stratified random sampling with two auxiliary variables under double sampling design. Theoretical and empirical results show that the proposed estimator is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Chand (1975) chain ratio estimator, Choudhary and Singh (2012) estimator, chain ratio-product-type estimator, Sahoo et al. (1993) difference type estimator, and Kiregyera (1984) regression-type estimator. Two data sets are used to illustrate the performances of different estimators.     

Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response

In this paper, bias-adjustment in the jackknife estimator of variance accredited to Rao and Sitter (1995) has been considered. Then the bias-adjusted Rao and Sitter (1995) estimator has been calibrated such that its expected value under the imputing superpopulation model remains the same as the expected value of the mean squared error of the ratio estimator in the presence of non-response. A simulation study has been performed to compare the six different estimators of variance: out of them four estimators belong to Rao and Sitter (1995) and the other two proposed estimators are named as bias-adjusted and bias-adjusted-cum-calibrated estimators. The empirical relative bias and empirical relative efficiency of the two proposed estimators with respect to the four existing estimators accredited to Rao and Sitter (1995) have been investigated through simulations. The bias-adjusted-cum-calibrated estimator has been found to be an efficient estimator in the case of heteroscadastic populations. The present paper considers the situation of simple random and without replacement sampling. The possibility of obtaining a negative estimate of variance by the estimator due to Kim et al. (2006) has been pointed out.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

Separate Ratio Estimators for the Population Variance in Stratified Random Sampling

We propose separate ratio estimators for population variance in stratified random sampling. We obtain mean square error equations and compare proposed estimators about efficiency with each other. By these comparisons, we find the conditions which make proposed estimators more efficient than others. It has been shown that proposed classes of estimators are more efficient than usual unbiased estimator. We find that separate ratio estimators are more efficient than combined ratio estimators for population variance. The theoretical results are supported by a numerical illustration with original data. A simulation study is also carried out to investigate empirical performance of estimators.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

A note on the estimation of variance of sample mean Using the knowledge of coefficient of variation in normal population

A class of estimators for the variance of sample mean is defined and its properties are studied in case of normal population. It is identified that the usual unbiased estimator, Singh, Pandey and Hirano (1973) -type estimator and Lee (1931) estimator are particular members of the proposed class of estimators. It is found that the minimum Mean Squared Error (MSE) of the proposed class of estimators is less than that of other estimators.     

A study on the chain ratio-ratio-type exponential estimator for finite population variance

This paper considers the problem of estimating the population variance S²_y of the study variable y using the auxiliary information in sample surveys. We have suggested the (i) chain ratio-type estimator (on the lines of Kadilar and Cingi (2003)), (ii) chain ratio-ratio-type exponential estimator and their generalized version [on the lines of Singh and Pal (2015)] and studied their properties under large sample approximation. Conditions are obtained under which the proposed estimators are more efficient than usual unbiased estimator s²_y and Isaki (1893) ratio estimator. Improved version of the suggested class of estimators is also given along with its properties. An empirical study is carried out in support of the present study.     

MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

On the estimation of the true probability of a correct selection

For ranking and selection problems, the true probabiIity of a correct selection P(CS) is unknown even if a selection is made under the indifference-zone approach. Thus to estimate the true P(CS) some Bayes estimators and a bootstrap estimator are proposed for two normcal populations with common known variance. Also a bootstrap estimator and a bootstrap confidence interval are proposed for normal populations with common unknown variance. Some comparisons between proposed estimators and some other known estimators are made via Monte Carlo simulations.     

Using generalized doubly robust estimator to estimate average treatment effects of multiple treatments in observational studies

The generalized doubly robust estimator is proposed for estimating the average treatment effect (ATE) of multiple treatments based on the generalized propensity score (GPS). In medical researches where observational studies are conducted, estimations of ATEs are usually biased since the covariate distributions could be unbalanced among treatments. To overcome this problem, Imbens [The role of the propensity score in estimating dose-response functions, Biometrika 87 (2000), pp. 706–710] and Feng et al. [Generalized propensity score for estimating the average treatment effect of multiple treatments, Stat. Med. (2011), in press. Available at: http://onlinelibrary.wiley.com/doi/10.1002/sim.4168/abstract] proposed weighted estimators that are extensions of a ratio estimator based on GPS to estimate ATEs with multiple treatments. However, the ratio estimator always produces a larger empirical sample variance than the doubly robust estimator, which estimates an ATE between two treatments based on the estimated propensity score (PS). We conduct a simulation study to compare the performance of our proposed estimator with Imbens’ and Feng et al.’s estimators, and simulation results show that our proposed estimator outperforms their estimators in terms of bias, empirical sample variance and mean-squared error of the estimated ATEs.     

Small area estimation of mean price of habitation transaction using time-series and cross-sectional area-level models

In this paper, a new small domain estimator for area-level data is proposed. The proposed estimator is driven by a real problem of estimating the mean price of habitation transaction at a regional level in a European country, using data collected from a longitudinal survey conducted by a national statistical office. At the desired level of inference, it is not possible to provide accurate direct estimates because the sample sizes in these domains are very small. An area-level model with a heterogeneous covariance structure of random effects assists the proposed combined estimator. This model is an extension of a model due to Fay and Herriot [5], but it integrates information across domains and over several periods of time. In addition, a modified method of estimation of variance components for time-series and cross-sectional area-level models is proposed by including the design weights. A Monte Carlo simulation, based on real data, is conducted to investigate the performance of the proposed estimators in comparison with other estimators frequently used in small area estimation problems. In particular, we compare the performance of these estimators with the estimator based on the Rao–Yu model [23]. The simulation study also accesses the performance of the modified variance component estimators in comparison with the traditional ANOVA method. Simulation results show that the estimators proposed perform better than the other estimators in terms of both precision and bias.     

A Class of Estimators of Population Variance Using Auxiliary Information in Sample Surveys

This article advocates the problem of estimating the population variance of the study variable using information on certain known parameters of an auxiliary variable. A class of estimators for population variance using information on an auxiliary variable has been defined. In addition to many estimators, usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999), and Kadilar and Cingi's (2006) estimators are shown as members of the proposed class of estimators. Asymptotic expressions for bias and mean square error of the proposed class of estimators have been obtained. An empirical study has been carried out to judge the performance of the various estimators of population variance generated from the proposed class of estimators over usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999) and Kadilar and Cingi's (2006) estimators.     

Second-order variance estimation in poststratified two-stage sampling

The linearization or Taylor series variance estimator and jackknife linearization variance estimator are popular for poststratified point estimators. In this note we propose a simple second-order linearization variance estimator for the poststratified estimator of the population total in two-stage sampling, using the second-order Taylor series expansion. We investigate the properties of the proposed variance estimator and its modified version and their empirical performance through some simulation studies in comparison to the standard and jackknife linearization variance estimators. Simulation studies are carried out on both artificially generated data and real data.     

Some unbiased estimators versus mean per unit and ratio estimators in finite population sample surveys

We present some unbiased estimators at the population mean in a finite population sample surveys with simple random sampling design where information on an auxiliary variance x positively correlated with the main variate y is available. Exact variance and unbiased estimate of the variance are computed for any sample size. These estimators are compared for their precision with the mean per unit and the ratio estimators. Modifications of the estimators are suggested to make them more precise than the mean per unit estimator or the ratio estimator regardless of the value of the population correlation coefficient between the variates x and y. Asymptotic distribution of our estimators and confidnece intervals for the population mean are also obtained.     

Efficient use of multi-auxiliary information in search of good rotation patterns in successive sampling

This article considers the problem of estimating the population mean on the current (second) occasion using multi-auxiliary information in successive sampling over two occasions. A general class of estimators is proposed for estimating population mean on the current occasion and expressions for bias and mean square error for these estimators are obtained up to first degree of approximation. The minimum variance bound estimator in the proposed class is discussed. Many popular estimators have been shown to belong to this class. Optimum replacement policy is also discussed. Finally, the superiority of the proposed class of estimators over multivariate version of chain type ratio estimator envisaged by Singh (2005 Singh, G.N. (2005). On the use of chain type ratio estimator in successive sampling. Stat Transition 7:21–26. [Google Scholar]) is established empirically.     

Chain ratio type estimator for ratio of two population m3ans using auxiliary characters

In this paper we have proposed chain ratio type estimators for ratio of two population means using two auxiliary characters. The expressions for bias and mean square error of these estimators have been derived. A comparison of the proposed estimator with that of double sampling estimator has been made in terms of mean square error. An emperical study has also been made.     

Estimating the variance in nonparametric regression—what is a reasonable choice?

The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titterington's optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rice's estimator does not always behave well. Data-driven guidelines are given to select the estimator with the smallest MSE.     

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.