Efficient estimators for adaptive stratified sequential sampling

In stratified sampling, methods for the allocation of effort among strata usually rely on some measure of within-stratum variance. If we do not have enough information about these variances, adaptive allocation can be used. In adaptive allocation designs, surveys are conducted in two phases. Information from the first phase is used to allocate the remaining units among the strata in the second phase. Brown et al. [Adaptive two-stage sequential sampling, Popul. Ecol. 50 (2008), pp. 239–245] introduced an adaptive allocation sampling design – where the final sample size was random – and an unbiased estimator. Here, we derive an unbiased variance estimator for the design, and consider a related design where the final sample size is fixed. Having a fixed final sample size can make survey-planning easier. We introduce a biased Horvitz–Thompson type estimator and a biased sample mean type estimator for the sampling designs. We conduct two simulation studies on honey producers in Kurdistan and synthetic zirconium distribution in a region on the moon. Results show that the introduced estimators are more efficient than the available estimators for both variable and fixed sample size designs, and the conventional unbiased estimator of stratified simple random sampling design. In order to evaluate efficiencies of the introduced designs and their estimator furthermore, we first review some well-known adaptive allocation designs and compare their estimator with the introduced estimators. Simulation results show that the introduced estimators are more efficient than available estimators of these well-known adaptive allocation designs.     

An adaptive allocation sampling design for which the conventional stratified estimator is an appropriate estimator

Adaptive allocations in stratified sampling design are suitable for studying Biological and Environmental populations. Biologists tend to use the conventional stratified estimator for an adaptive allocation sampling design in spite of its adaptive nature. In this paper, we have introduced an adaptive allocation sampling design for which the conventional stratified estimator has some desired statistical properties. We conduct a simulation study on a real population. The results show that the conventional stratified estimator for the introduced adaptive allocation sampling design is more efficient than other available estimators.     

Using adaptive line-transect sampling in airborne geophysics studies

We introduce a design that combines elements from distance and adaptive cluster sampling designs. We propose a line-transect sampling method, where the sample-strips are selected by unequal selection probabilities, detectability of clusters is assumed imperfect and detectability of sample units belonging to each detected cluster is assumed perfect. Here, the application of distance sampling is broaden to airborne geophysics studies. We introduce efficient estimators for this new sample design. Also, we conduct two simulation studies. One of the populations is Sar Cheshmeh Copper mine with data from an airborne geophysics study and the other is an artificial population.     

Unbiased estimators for restricted adaptive cluster sampling

In adaptive cluster sampling the size of the final sample is random, thus creating design problems. To get round this, Brown (1994) and Brown & Manly (1998) proposed a modification of the method, placing a restriction on the size of the sample, and using standard but biased estimators for estimating the population mean. But in this paper a new unbiased estimator and an unbiased variance estimator are proposed, based on estimators proposed by Murthy (1957) and extended to sequential and adaptive sampling designs by Salehi & Seber (2001). The paper also considers a restricted version of the adaptive scheme of Salehi & Seber (1997a) in which the networks are selected without replacement, and obtains unbiased estimators. The method is demonstrated by a simple example. Using simulation from this example, the new estimators are shown to compare very favourably with the standard biased estimators.     

A general inverse sampling scheme and its application to adaptive cluster sampling

Not having a variance estimator is a seriously weak point of a sampling design from a practical perspective. This paper provides unbiased variance estimators for several sampling designs based on inverse sampling, both with and without an adaptive component. It proposes a new design, which is called the general inverse sampling design, that avoids sampling an infeasibly large number of units. The paper provide estimators for this design as well as its adaptive modification. A simple artificial example is used to demonstrate the computations. The adaptive and non‐adaptive designs are compared using simulations based on real data sets. The results indicate that, for appropriate populations, the adaptive version can have a substantial variance reduction compared with the non‐adaptive version. Also, adaptive general inverse sampling with a limitation on the initial sample size has a greater variance reduction than without the limitation.     

Estimation of rare and clustered population variance in adaptive cluster sampling

Abstract

Many researchers used auxiliary information together with survey variable to improve the efficiency of population parameters like mean, variance, total and proportion. Ratio and regression estimation are the most commonly used methods that utilized auxiliary information in different ways to get the maximum benefits in the form of high precision of the estimators. Thompson first introduced the concept of Adaptive cluster sampling, which is an appropriate technique for collecting the samples from rare and clustered populations. In this article, a generalized exponential type estimator is proposed and its properties have been studied for the estimation of rare and highly clustered population variance using single auxiliary information. A numerical study is carried out on a real and artificial population to judge the performance of the proposed estimator over the competing estimators. It is shown that the proposed generalized exponential type estimator is more efficient than the adaptive and non adaptive estimators under conventional sampling design.     

Some calibration estimators for finite population mean in two-stage stratified random sampling

Abstract

In the present article, an effort has been made to develop calibration estimators of the population mean under two-stage stratified random sampling design when auxiliary information is available at primary stage unit (psu) level. The properties of the developed estimators are derived in-terms of design based approximate variance and approximate consistent design based estimator of the variance. Some simulation studies have been conducted to investigate the relative performance of calibration estimator over the usual estimator of the population mean without using auxiliary information in two-stage stratified random sampling. Proposed calibration estimators have outperformed the usual estimator without using auxiliary information.     

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.     

A New Proof of Murthy's Estimator which Applies to Sequential Sampling

Murthy's estimator has been used for constructing an unbiased estimator of a population total or mean from a sample of fixed size when there is unequal probability sampling without replacement. Traditionally, the estimator is derived by constructing an unordered version of Raj's ordered unbiased estimator. This paper presents an elementary proof of Murthy's estimator which applies the Rao–Blackwell theorem to a very simple estimator. This proof includes any sequential sampling scheme, thus extending the usefulness of Murthy's estimator. We demonstrate this extension by deriving unbiased estimators for inverse sampling.     

Variance estimation of DEE and regression estimator when a first-phase cluster sample is restratified

We consider a variance estimation when a stratified single stage cluster sample is selected in the first phase and a stratified simple random element sample is selected in the second phase. We propose explicit formulas of (asymptotically), we propose explicit formulas of (asymptotically) unbiased variance estimators for the double expansion estimator and regression estimator. We perform a small simulation study to investigate the performance of the proposed variance estimators. In our simulation study, the proposed variance estimator showed better or comparable performance to the Jackknife variance estimator. We also extend the results to a two-phase sampling design in which a stratified pps with replacement cluster sample is selected in the first phase.     

Mixed ranked set sampling design

The main focus of agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one method that leads to accomplish such objectives by incorporating expert knowledge to its advantage. In this paper, we propose an efficient sampling scheme, named mixed RSS (MxRSS), for estimation of the population mean and median. The MxRSS scheme is a suitable mixture of both simple random sampling (SRS) and RSS schemes. The MxRSS scheme provides an unbiased estimator of the population mean, and its variance is always less than the variance of sample mean based on SRS. For both symmetric and asymmetric populations, the mean and median estimators based on SRS, partial RSS (PRSS) and MxRSS schemes are compared. It turns out that the mean and median estimates under MxRSS scheme are more precise than those based on SRS scheme. Moreover, when estimating the mean of symmetric and some asymmetric populations, the mean estimates under MxRSS scheme are found to be more efficient than the mean estimates with PRSS scheme. An application to real data is also provided to illustrate the implementation of the proposed sampling scheme.     

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.     

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.     

Stein-type estimation using ranked set sampling

In this study, we consider the application of the James–Stein estimator for population means from a class of arbitrary populations based on ranked set sample (RSS). We consider a basis for optimally combining sample information from several data sources. We succinctly develop the asymptotic theory of simultaneous estimation of several means for differing replications based on the well-defined shrinkage principle. We showcase that a shrinkage-type estimator will have, under quadratic loss, a substantial risk reduction relative to the classical estimator based on simple random sample and RSS. Asymptotic distributional quadratic biases and risks of the shrinkage estimators are derived and compared with those of the classical estimator. A simulation study is used to support the asymptotic result. An over-riding theme of this study is that the shrinkage estimation method provides a powerful extension of its traditional counterpart for non-normal populations. Finally, we will use a real data set to illustrate the computation of the proposed estimators.     

A balanced multi-level rotation sampling design and its efficient composite estimators

We present a multi-level rotation sampling design which includes most of the existing rotation designs as special cases. When an estimator is defined under this sampling design, its variance and bias remain the same over survey months, but it is not so under other existing rotation designs. Using the properties of this multi-level rotation design, we derive the mean squared error (MSE) of the generalized composite estimator (GCE), incorporating the two types of correlations arising from rotating sample units. We show that the MSEs of other existing composite estimators currently used can be expressed as special cases of the GCE. Furthermore, since the coefficients of the GCE are unknown and difficult to determine, we present the minimum risk window estimator (MRWE) as an alternative estimator. This MRWE has the smallest MSE under this rotation design and yet, it is easy to calculate. The MRWE is unbiased for monthly and yearly changes and preserves the internal consistency in total. Our numerical study shows that the MRWE is as efficient as GCE and more efficient than the existing composite estimators and does not suffer from the drift problem [Fuller W.A., Rao J.N.K., 2001. A regression composite estimator with application to the Canadian Labour Force Survey. Surv. Methodol. 27 (2001) 45–51] unlike the regression composite estimators.     

Confidence regions based on inefficient estimators

We consider an unbiased estimator of a function of mean value parameters, which is not efficient. This inefficient estimator is correlated with a residual vector. Thus, if a unit dispersion is unknown, it is impossible to determine the correct confidence region for a function of mean value parameters via a standard estimator of an unknown dispersion with the exception of the case when the ordinary least squares (OLS) estimator is considered in a model with a special covariance structure such that the OLS and the generalized least squares (GLS) estimator are the same, that is the OLS estimator is efficient. Two different estimators of a unit dispersion independent of an inefficient estimator are derived in a singular linear statistical model. Their quality was verified by simulations for several types of experimental designs. Two new estimators of the unit dispersion were compared with the standard estimators based on the GLS and the OLS estimators of the function of the mean value parameters. The OLS estimator was considered in the incorrect model with a different covariance matrix such that the originally inefficient estimator became efficient. The numerical examples led to a slightly surprising result which seems to be due to data behaviour. An example from geodetic practice is presented in the paper.     

Performance of nonparametric maximum likelihood estimator in a

When the probability of selecting an individual in a population is propor­tional to its lifelength, it is called length biased sampling. A nonparametric maximum likelihood estimator (NPMLE) of survival in a length biased sam­ple is given in Vardi (1982). In this study, we examine the performance of Vardi's NPMLE in estimating the true survival curve when observations are from a length biased sample. We also compute estimators based on a linear combination (LCE) of empirical distribution function (EDF) estimators and weighted estimators. In our simulations, we consider observations from a mix­ture of two different distributions, one from F and the other from G which is a length biased distribution of F. Through a series of simulations with vari­ous proportions of length biasing in a sample, we show that the NPMLE and the LCE closely approximate the true survival curve. Throughout the sur­vival curve, the EDF estimators overestimate the survival. We also consider a case where the observations are from three different weighted distributions, Again, both the NPMLE and the LCE closely approximate the true distribu­tion, indicating that the length biasedness is properly adjusted for. Finally, an efficiency study shows that Vardi's estimators are more efficient than the EDF estimators in the lower percentiles of the survival curves.     

Two-Stage Cluster Sampling with Ranked Set Sampling in the Secondary Sampling Frame

Motivated by a real-life problem, we develop a Two-Stage Cluster Sampling with Ranked Set Sampling (TSCRSS) design in the second stage for which we derive an unbiased estimator of population mean and its variance. An unbiased estimator of the variance of mean estimator is also derived. It is proved that the TSCRSS is more efficient—in the sense of having smaller variance—than the conventional two-stage cluster simple random sampling in which the second-stage sampling is with replacement. Using a simulation study on a real-life population, we show that the TSCRSS is more efficient than the conventional two-stage cluster sampling when simple random sampling without replacement is used in both stages.     

Generalized ratio-type and ratio-exponential-type estimators for population mean under modified Horvitz-Thompson estimator in adaptive cluster sampling

In the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, mid-range) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.     

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.