Generalized estimator for the estimation of rare and clustered population variance in adaptive cluster sampling

Adaptive cluster sampling (ACS) is considered to be the most suitable sampling design for the estimation of rare, hidden, clustered and hard-to-reach population units. The main characteristic of this design is that it may select more meaningful samples and provide more efficient estimates for the field investigator as compare to the other conventional sampling designs. In this paper, we proposed a generalized estimator with a single auxiliary variable for the estimation of rare, hidden and highly clustered population variance under ACS design. The expressions of approximate bias and mean square error are derived and the efficiency comparisons have been made with other existing estimators. A numerical study is carried out on a real population of aquatic birds together with an artificial population generated by Poisson cluster process. Related results of numerical study show that the proposed generalized variance estimator is able to provide considerably better results over the competing estimators.     

On the consistency between model-and design-based estimators in survey sampling

In finite population sampling, often a distinction is made between model-and design-based estimators of the parameters of interest (like the population total, population variance, etc.). The model-based estimators depend on the (known) parameters of the model, while the design-based estimators depend on the (known) selection probabilities of the different units in the population. It is shown in this paper that the two approaches are not necessarily incompatible, and indeed can often lead to the same estimator. Our ideas are illustrated with the Horvitz-Thompson, and the generalized Horvitz-Thompson estimator. These estimators are identified as hierarchical Bays estimators. Also, certain “stepwise-Bayes” estimators of Vardeman and Meeden (J. Stat. Inf. (1983), V7, pp 329-341) are unified from a hierarchical Bayes point of view.     

Estimation of Population Mean Using Known Coefficient of Variation of the Study Character in the Presence of Non Response

Using the known coefficient of variation of the study character, generalized and regression-type estimators for the population mean using two phase sampling in the presence of non response were proposed and their properties have been studied. The conditions under which the proposed estimators are more efficient than the relevant estimators have been obtained. The empirical studies were given in the support of the problems in the case of positive and negative correlation between the study and the auxiliary characters which show the increase in the efficiency of the proposed estimators using known coefficient of variation of the study character with respect to the relevant estimators.     

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.     

Use of scrambled response model in estimating the finite population mean in presence of non response when coefficient of variation is known

We propose an estimator for the finite population mean utilizing known coefficient of variation of the study character in case of quantitative sensitive variable considering a randomization mechanism on the second call that provides privacy protection to the respondents to get truthful information. We also propose generalized ratio- and regression-type estimators under two-phase sampling scheme. The conditions under which the proposed estimators are more efficient than the relevant estimators under scrambled response model have been obtained. An empirical study is carried out to evaluate performances of the estimators.     

ON NON-NEGATIVE UNBIASED ESTIMATION OF QUADRATIC FORMS IN FINITE POPULATION SAMPLING

The paper investigates non-negative quadratic unbiased (NnQU) estimators of positive semi-definite quadratic forms, for use during the survey sampling of finite population values. It examines several different NnQU estimators of the variance of estimators of population total, under various sampling designs. It identifies an optimal quadratic unbiased estimator of the variance of the Horvitz-Thompson estimator of population total.     

MODEL-UNBIASEDNESS AND THE HORVITZ-THOMPSON ESTIMATOR IN FINITE POPULATION SAMPLING

Under the, notion of superpopulation models, the concept of minimum expected variance is adopted as an optimality criterion for design-unbiased estimators, i.e. unbiased under repeated sampling. In this article, it is shown that the Horvitz-Thompson estimator is optimal among such estimators if and only if it is model-unbiased, i.e. unbiased under the model. The family of linear models is considered and a sample design is suggested to preserve the model-unbiasedness (and hence the optimality) of the Horvitz-Thompson estimator. It is also shown that under these models the Horvitz-Thompson estimator together with the suggested sample design is optimal among design-unbiased estimators with any sample design (of fixed size n ) having non-zero probabilities of inclusion for all population units.     

Mean square error comparison among variance estimators with known coefficient of variation

The improved estimators for the population parameters were considered by several statisticians under various conditions. Recently Laheetharan and Wijekoon (Improved estimation of the population parameters when some additional information is available. Stat Papers doi:, 2008) demonstrated a generalized procedure for obtaining optimal shrunken estimators, and derived such estimators for both population mean and variance when coefficient of variation is known. In this article the mean square errors of those estimators were compared, and a numerical illustration was done using the scaled mean square error loss as used by Kanefuji and Iwase (Stat Papers 39:377–388, 1998) to understand the efficiency of the estimators with increasing sample size.     

A Generalized Class of Ratio Type Exponential Estimators of Population Mean Under Linear Transformation of Auxiliary Variable

Recently, Shabbir and Gupta [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] defined a class of ratio type exponential estimators of population mean under a very specific linear transformation of auxiliary variable. In the present article, we propose a generalized class of ratio type exponential estimators of population mean in simple random sampling under a very general linear transformation of auxiliary variable. Shabbir and Gupta's [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] class of estimators is a particular member of our proposed class of estimators. It has been found that the optimal estimator of our proposed generalized class of estimators is always more efficient than almost all the existing estimators defined under the same situations. Moreover, in comparison to a few existing estimators, our proposed estimator becomes more efficient under some simple conditions. Theoretical results obtained in the article have been verified by taking a numerical illustration. Finally, a simulation study has been carried out to see the relative performance of our proposed estimator with respect to some existing estimators which are less efficient under certain conditions as compared to the proposed estimator.     

A generalized class of estimators under two-phase stratified sampling for non response

In this paper, we propose a generalized class of estimators for finite population mean using two auxiliary variables in two-phase stratified sampling for non response. We identify 17 estimators as special cases of the proposed class of estimators. Expressions for the bias and mean squared error (MSE) of estimators are obtained up to first order of approximation. A data set is used for efficiency comparisons.     

On optimality of the horvitz-thompson estimator under a markov process model

For any varying probability sampling design the Horvitz-Thompson (1952) estimator is shown to be optimal within the class of all unbiased estimators of a finite population total under a Markov process model     

Improved two phase sampling exponential ratio and product type estimators for population mean of study character in the presence of non response

Improved two phase sampling exponential ratio and product type estimators for population mean using known coefficient of variation of study character in the presence of non response have been proposed and their properties are studied under large sample approximation. The proposed estimators are compared with the other existing estimators by using the MSE criterion and the conditions under which the proposed estimators perform better are obtained. An empirical study is also given to judge the performance of the proposed estimators. At the end, simulation studies have been carried out to verify the superiority to the proposed estimators.     

Recursive computation of inclusion probabilities in ranked-set sampling

We derive recursive algorithms for computing first-order and second-order inclusion probabilities for ranked-set sampling from a finite population. These algorithms make it practical to compute inclusion probabilities even for relatively large sample and population sizes. As an application, we use the inclusion probabilities to examine the performance of Horvitz-Thompson estimators under different varieties of balanced ranked-set sampling. We find that it is only for balanced Level 2 sampling that the Horvitz-Thompson estimator can be relied upon to outperform the simple random sampling mean estimator.     

An Improved Generalized Difference-Cum-Ratio-Type Estimator for the Population Variance in Two-Phase Sampling Using Two Auxiliary Variables

In this paper, an improved generalized difference-cum-ratio-type estimator for the finite population variance under two-phase sampling design is proposed. The expressions for bias and mean square error (MSE) are derived to first order of approximation. The proposed estimator is more efficient than the usual sample variance estimator, traditional ratio estimator, traditional regression estimator, chain ratio type and chain ratio-product-type estimators, and Jhajj and Walia (2011) estimator. Four datasets are also used to illustrate the performances of different estimators.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Survey sampling in graphs

The Horvitz-Thompson estimation theory is applied to snowball sampling and some other sampling procedures using a known or unknown graph structure in the survey population. In particular, simple graph-parameter estimators and variance estimators are obtained which are based on various kinds of partial information about the graph.     

Estimation of population coefficient of variation in simple and stratified random sampling under two-phase sampling scheme when using two auxiliary variables

We propose an improved class of exponential ratio type estimators for coefficient of variation (CV) of a finite population in simple and stratified random sampling using two auxiliary variables under two-phase sampling scheme. We examine the properties of the proposed estimators based on first order of approximation. The proposed class of estimators is more efficient than the usual sample CV estimator, ratio estimator, exponential ratio estimator, usual difference estimator and modified difference type estimator. We also use real data sets for numerical comparisons.     

Optimized estimation for population mean using conventional and non-conventional measures under the joint influence of measurement error and non-response

Most of the research work in the theory of survey sampling only deals with the sampling errors under the assumptions: (i) there is a complete response and (ii) recorded information from individuals is correct but in practice it is not always true. Non-sampling errors like non-response and measurement errors (MEs) mostly creep into the survey and become more influential for estimators than sampling errors. Considering this practical situation of non-response and MEs jointly, we proposed an optimum class of estimators for population mean under simple random sampling using conventional and non-conventional measures. Bias and mean square error of the proposed estimators are derived up to first degree of approximation. Moreover, a simulation study is conducted to assess the performance of new estimators which proves that proposed estimators are more efficient than the traditional Hansen and Hurwitz estimator and other competing estimators.