New calibration estimator based on two auxiliary variables in stratified sampling

In recent years, calibration estimation has become an important field of research in survey sampling. This paper proposes a new calibration estimator for the population mean in the presence of two auxiliary variables in stratified sampling. The theory of new calibration estimator is given and optimum calibration weights are derived. A simulation study is carried out to performance of the proposed calibration estimator over other existing calibration estimators. The results reveal that the proposed calibration estimators are more efficient than other existing calibration estimators in stratified sampling.     

New calibration estimator in stratified sampling

This paper develops the theory of calibration estimation and proposes calibration approach alternative to existing calibration estimators for estimating population mean of the study variable using auxiliary variable in stratified sampling. The theory of new calibration estimation is given and optimum weights are derived. A simulation study is carried out to performance of the proposed calibration estimator with other existing calibration estimators. The results reveal that the proposed calibration estimators are more efficient than Tracy et al., Singh et al., Singh calibration estimators of the population mean.     

基于多重抽样框的校准估计方法研究

在现代抽样调查中,校准估计方法能够通过有效利用辅助信息来提高估计量的精度,多重抽样框抽样调查则不仅可以解决单一抽样框覆盖不全的问题,还可以节约抽样设计阶段的成本。本文将这两种现代抽样估计与设计方法进行结合,将校准估计方法引入到基于多重抽样框的抽样调查体系中,在实现节约调查成本的同时,还能够提高估计量的精度。文章首先按照分离抽样框与组合抽样框估计方法的分类思路,对传统多重抽样框估计方法进行系统梳理;然后在最短距离法校准估计的分析框架下,按照调查时所能掌握辅助信息的具体情况,给出了两类多重抽样框估计情形下的各种不同形式的校准估计量;随后数值分析的比较结果也表明在多重抽样框中校准估计量的估计效率明显优于传统估计量;最后对本文研究进行总结的基础上,给出了我国抽样实践中应用这套先进抽样估计方法体系的展望。     

A note on calibration weightings for stratified double sampling with equal probability

This study proposes a more efficient calibration estimator for estimating population mean in stratified double sampling using new calibration weights. The variance of the proposed calibration estimator has been derived under large sample approximation. Calibration asymptotic optimum estimator and its approximate variance estimator are derived for the proposed calibration estimator and existing calibration estimators in stratified double sampling. Analytical results showed that the proposed calibration estimator is more efficient than existing members of its class in stratified double sampling. Analysis and evaluation are presented.     

Calibrated estimators in two-stage sampling

We consider the problem of the estimation of the population mean of a study variable by assuming that the population means of an auxiliary variable are known at both stages of sample selection. The design weights at the first and second stages of sample selection are calibrated by optimizing the chi-squared type distance between the design weights and the new weights at both the first and second stages of sample selection. The regression type estimator in two-stage sampling is shown to be a special case. An application of the proposed estimators using a real data set is discussed.     

Calibration Weighting in Stratified Random Sampling

A new calibration estimator is proposed to estimate the population mean in the stratified random sampling. The corrected expression of Tracy et al. (2003) calibrated weights are presented and new improved calibration weights are introduced. Theoretical variance of the suggested estimator is discussed. Also a simulation study is carried out to show the properties of the proposed estimator.     

Two-step calibration of design weights under two auxiliary variables in sample survey

Calibration on the available auxiliary variables is widely used to increase the precision of the estimates of parameters. Singh and Sedory [Two-step calibration of design weights in survey sampling. Commun Stat Theory Methods. 2016;45(12):3510–3523.] considered the problem of calibration of design weights under two-step for single auxiliary variable. For a given sample, design weights and calibrated weights are set proportional to each other, in the first step. While, in the second step, the value of proportionality constant is determined on the basis of objectives of individual investigator/user for, for example, to get minimum mean squared error or reduction of bias. In this paper, we have suggested to use two auxiliary variables for two-step calibration of the design weights and compared the results with single auxiliary variable for different sample sizes based on simulated and real-life data set. The simulated and real-life application results show that two-auxiliary variables based two-step calibration estimator outperforms the estimator under single auxiliary variable in terms of minimum mean squared error.     

Best linear unbiased estimates in ranked-set sampling with particular reference to imperfect ordering

SUMMARY Ranked-set sampling is a widely used sampling procedure when sample observations are expensive or difficult to obtain. It departs from simple random sampling by seeking to spread the observations in the sample widely over the distribution or population. This is achieved by ranking methods which may need to employ concomitant information. The ranked-set sample mean is known to be more efficient than the corresponding simple random sample mean. Instead of the ranked-set sample mean, this paper considers the corresponding optimal estimator: the ranked-set best linear unbiased estimator. This is shown to be more efficient, even for normal data, but particularly for skew data, such as from an exponential distribution. The corresponding forms of the estimators are quite distinct from the ranked-set sample mean. Improvement holds where the ordering is perfect or imperfect, with this prospect of improper ordering being explored through the use of concomitants. In addition, the corresponding optimal linear estimator of a scale parameter is also discussed. The results are applied to a biological problem that involves the estimation of root weights for experimental plants, where the expense of measurement implies the need to minimize the number of observations taken.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

Rare and clustered population estimation using the adaptive cluster sampling with some robust measures

The use of robust measures helps to increase the precision of the estimators, especially for the estimation of extremely skewed distributions. In this article, a generalized ratio estimator is proposed by using some robust measures with single auxiliary variable under the adaptive cluster sampling (ACS) design. We have incorporated tri-mean (TM), mid-range (MR) and Hodges-Lehman (HL) of the auxiliary variable as robust measures together with some conventional measures. The expressions of bias and mean square error (MSE) of the proposed generalized ratio estimator are derived. Two types of numerical study have been conducted using artificial clustered population and real data application to examine the performance of the proposed estimator over the usual mean per unit estimator under simple random sampling (SRS). Related results of the simulation study show that the proposed estimators provide better estimation results on both real and artificial population over the competing estimators.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

New simulation results for the calibration and inverse median estimation problems

This paper gives the results of a new simulation study for the familiar calibration problem and the less familiar inverse median estimation problem. The latter arises when one wishes to estimate from a linear regression analysis the value of the independent variable corresponding to a specified value of the median of the dependent variable. For example, from the results of a regression analysis between stress and time to failure, one might wish to estimate the stress at which the median time to failure is 10,000 hours. In the study, the mean square error, Pitman closeness, and probability of overestimation are compared for both the calibration problem and the inverse median estimation problem for (1) the classical estimator, (2) the inverse estimator, and (3) a modified version of an estimator proposed by Naszodi (1978) for both a small sample and a moderately large sample situation.     

Consistency of the recursive nonparametric regression estimation for dependent functional data

We consider the recursive estimation of a regression functional where the explanatory variables take values in some functional space. We prove the almost sure convergence of such estimates for dependent functional data. Also we derive the mean quadratic error of the considered class of estimators. Our results are established with rates and asymptotic appear bounds, under strong mixing condition. Finally, the feasibility of the proposed estimator is illustrated throughout an empirical study.     

A dual problem of calibration of design weights

In this note, a dual problem to the calibration of design weights of the Deville and Särndal [Calibration estimators in survey sampling, J. Amer. Statist. Assoc. 87 (1992), pp. 376–382] method has been considered. We conclude that the chi-squared distance between the design weights and the calibrated weights equals the square of the standardized Z-score obtained by the difference between the known population total of the auxiliary variable and its corresponding Horvitz and Thompson [A generalization of sampling without replacement from a finite universe, J. Amer. Statist. Assoc. 47 (1952), pp. 663–685] estimator divided by the sample standard deviation of the auxiliary variable to obtain the linear regression estimator in survey sampling.     

A conditional perspective of weighted variance estimation of the optimal regression estimator

The estimation of the variance for the GREG (general regression) estimator by weighted residuals is widely accepted as a method which yields estimators with good conditional properties. Since the optimal (regression) estimator shares the properties of GREG estimators which are used in the construction of weighted variance estimators, we introduce the weighting procedure also for estimating the variance of the optimal estimator. This method of variance estimation was originally presented in a seemingly ad hoc manner, and we shall discuss it from a conditional point of view and also look at an alternative way of utilizing the weights. Examples that stress conditional behaviour of estimators are then given for elementary sampling designs such as simple random sampling, stratified simple random sampling and Poisson sampling, where for the latter design we have conducted a small simulation study.     

On stratified bivariate ranked set sampling with optimal allocation for naïve and ratio estimators

The purpose of the current work is to introduce stratified bivariate ranked set sampling (SBVRSS) and investigate its performance for estimating the population mean using both naïve and ratio methods. The properties of the proposed estimator are derived along with the optimal allocation with respect to stratification. We conduct a simulation study to demonstrate the relative efficiency of SBVRSS as compared to stratified bivariate simple random sampling (SBVSRS) for ratio estimation. Data that consist of weights and bilirubin levels in the blood of 120 babies are used to illustrate the procedure on a real data set. Based on our simulation, SBVRSS for ratio estimation is more efficient than using SBVSRS in all cases.     

Calibration Approach-based Product Estimator of Finite Population Total with Subsampling of Nonrespondents under Single and Two-phase Sampling

Hansen and Hurwitz (1946) technique–based estimator of population total is proposed using the calibration approach under the assumption that the auxiliary variable is negatively correlated with the study variable. The variance estimation is also considered. The two-phase sampling case is also explored. The theoretical results are demonstrated through empirical studies using both generated and real population data. The proposed estimator of population total outperforms the existing estimators in terms of the criteria of relative bias and relative root mean square error.     

Linear functional regression: The case of fixed design and functional response

The authors consider the problem of simple linear regression when the exogenous and endogenous variables are functional and the design is fixed. They propose an estimator for the underlying linear operator and prove its consistency under some conditions which ensure that the design is sufficiently informative. They consider the classical calibration (or inverse regression) problem and analyze a consistent estimator. They also give a simulation study. The proposed method is not hard to implement in practice.     

Large-sample comparisons of calibration procedures when both measurements are subject to error: the unreplicated case

A predictive functional relationship model is presented for the calibration problem in which the standard as well as the nonstandard measurements are subject to error. For the estimation of the relationship between the two measurements, the ordinary least squares and maximum likelihood estimation methods are considered, while for the prediction of unknown standard measurements we consider direct and inverse approaches. Relative performances of those calibration procedures are compared in terms of the asymptotic mean square error of prediction.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.