Estimation of the number of vertices of different degrees in a graph

An unknown graph is partially observed by selecting a vertex sample and observing the edges in the subgraph induced by the sample. The sample is selected by either simple random sampling or Bernoulli sampling. We consider the problem of estimating the numbers of vertices of different degrees in the unknown graph by using the sample information. Unbiased estimators are given and their variance-covariance matrix is shown to depend on a set of intrinsic graph parameters which can hardly be satisfactorily estimated from the sample information without further assumptions. In particular, the problem of estimating the number of isolates (vertices of degree zero) is considered in some detail.     

Estimation of mean using double sampling for stratification and multivariate auxiliary information

Several estimators for estimating the mean of a principal variable are proposed based on double sampling for stratification (DSS) and multivariate auxiliary information. The general properties of the proposed estimators are studied, search for optimum estimators is made and the proposed estimators are compared with the corresponding estimators based on unstratified double sampling (USDS).     

Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables

In this article, we propose a new class of estimators to estimate the finite population mean by using two auxiliary variables under two different sampling schemes such as simple random sampling and stratified random sampling. The proposed class of estimators gives minimum mean squared error as compared to all other considered estimators. Some real data sets are used to observe the performances of the estimators. We show numerically that the proposed class of estimators performs better as compared to all other competitor estimators.     

A general procedure for estimating the mean using double sampling for stratification and multi-auxiliary information

This paper defines a general procedure for estimating the population mean of the study variate based on double sampling for stratification in presence of multi-auxiliary information. Classes of combined and separate estimators have been suggested and their properties are studied under large sample approximation. A class of unstratified double sampling estimators is also proposed with its properties. Asymptotic optimum estimators in the classes are identified with their approximate variance formulae. Further the proposed classes of estimators are compared with the corresponding class of estimators based on un-stratified double sampling. All findings are encouraging and support the soundness of the proposed procedure for mean estimation.     

Ranked set sampling with unequal samples for skew distributions

A ranked set sampling procedure with unequal samples for positively skew distributions (RSSUS) is proposed and used to estimate the population mean. The estimators based on RSSUS are compared with the estimators based on ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is observed that the relative precisions of the estimators based on RSSUS are higher than those of the estimators based on RSS and MRSS procedures.     

Estimation of parameters of 1-out-of-2:g repairable system

Moment estimators of l-out-of-2:G repairable system are supplied under four sampling schemes assuming that the failure and repair time distribution of the units are exponential with unknown parameters λ, μ respectively. Information metrices of the estimators are supplied. Also it is shown that the estimators are asymptotically normally distributed in every sampling scheme.     

Estimating variance components by using survey data

Summary. Inflation-type weighted estimators for variance components can be badly biased. Modified weighted estimators suggested in the literature are also badly biased for certain sampling designs. We propose new estimators for variance components, some of which are approximately unbiased regardless of the sampling design. These estimators require knowledge of the joint inclusion probabilities of the observations. The small sample properties of the estimators are studied via simulation for the simple one-way random-effects model. An application is given by using data from the US Hispanic Health and Nutrition Examination Survey.     

Multiple inverse sampling in post-stratification with subpopulation sizes unknown: a solution for quota sampling

We extend traditional inverse sampling to multiple case. We then modify the multiple inverse sampling design to a version with taking a simple random sample at the beginning similar to Chang et al (J. Statist. Plan. Inference 69 (1998) 209) and a truncated version similar to Chang et al (J. Statist. Plan. Inference 76 (1999) 215). Using Murthy (Sankhya 18 (1957) 379) we develop their unbiased estimators and their unbiased variance estimators. These unbiased estimators can also be applied to a frequently used sampling scheme called quota sampling by practitioners. The multiple inverse sampling may be viewed as an improved version of quota sampling in some sense. We show that our estimators for estimating the proportions (weights) of subpopulations are more efficient and robust than available estimators using a small simulation study.     

Weighting for unequal selection probabilities in multilevel models

When multilevel models are estimated from survey data derived using multistage sampling, unequal selection probabilities at any stage of sampling may induce bias in standard estimators, unless the sources of the unequal probabilities are fully controlled for in the covariates. This paper proposes alternative ways of weighting the estimation of a two-level model by using the reciprocals of the selection probabilities at each stage of sampling. Consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase. Scaling of the weights is proposed to improve the properties of the estimators and to simplify computation. Variance estimators are also proposed. In a limited simulation study the scaled weighted estimators are found to perform well, although non-negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small. The variance estimators perform extremely well. The procedures are illustrated using data from the survey of psychiatric morbidity.     

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.     

Estimating the population proportion in modified ranked set sampling methods

In this paper, proportion estimators and associated variance estimators are proposed for a binary variable with a concomitant variable based on modified ranked set sampling methods, which are extreme ranked set sampling (ERSS), median ranked set sampling (MRSS), percentile ranked set sampling (Per-RSS) and L ranked set sampling (LRSS) methods. The Monte Carlo simulation study is performed to compare the performance of the estimators based on bias, mean squared error, and relative efficiency for different levels of correlation coefficient, set and cycle sizes under normal and log-normal distributions. Moreover, the study is supported with real data application.     

Synthetic and composite estimation under a superpopulation model

Under a simple superpopulation model for an arbitrary sampling design we derive optimal linear unbiased estimators/predictors of a mean in a domain. They can be viewed as synthetic and composite estimators of small area estimation theory when no auxiliary variable is available. Moreover, we show that the only requirement for optimality of a sampling strategy is to use any sampling plan of fixed sample size together with traditional estimators (as designed for simple random sampling without replacement). Finally, for symmetric sampling plans, simplified formulas (based on the first two moments of sample sizes) for optimal synthetic and composite estimators and their MSE’s are derived. Throughout the paper we consistently use the model-design setup.     

Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution

This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. depending on the parameters and the probability of selecting a question. We showed that our proposed methods have better efficiencies than Land et al.’s randomized response model in some conditions. When the sizes of stratified populations are not given, other estimators are suggested using a stratified double sampling. For the proportional allocation, the difference between two variances in the stratified sampling and the stratified double sampling is given with the known rare unrelated attribute.     

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.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

The Horvitz-Thompson Theorem as a Unifying Perspective for Probability Sampling: With Examples from Natural Resource Sampling

Courses in sampling often lack a coherent structure because many related sampling designs, estimators, variances, and variance estimators are presented as separate cases. The Horvitz-Thompson theorem offers a needed integrating perspective for teaching the methods and fundamental concepts of probability sampling. Development of basic concepts in sampling via this approach provides the student with tools to solve more complicated problems, and helps to avoid some common stumbling blocks of beginning students. Examples from natural resource sampling are provided to illustrate applications and insight gained from this approach.     

New two-stage sampling designs based on neoteric ranked set sampling

Neoteric ranked set sampling (NRSS) is a recently developed sampling plan, derived from the well-known ranked set sampling (RSS) scheme. It has already been proved that NRSS provides more efficient estimators for population mean and variance compared to RSS and other sampling designs based on ranked sets. In this work, we propose and evaluate the performance of some two-stage sampling designs based on NRSS. Five different sampling schemes are proposed. Through an extensive Monte Carlo simulation study, we verified that all proposed sampling designs outperform RSS, NRSS, and the original double RSS design, producing estimators for the population mean with a lower mean square error. Furthermore, as with NRSS, two-stage NRSS estimators present some bias for asymmetric distributions. We complement the study with a discussion on the relative performance of the proposed estimators. Moreover, an additional simulation based on data of the diameter and height of pine trees is presented.     

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.     

Optimal domain estimation under summation restriction

The domain estimators that do not sum up to the population total (estimated or known) are considered. In order to achieve their additivity, the theory of the general restriction (GR)-estimator [Knottnerus P., 2003. Sample Survey Theory: Some Pythagorean Perspectives. Springer, New York] is used. The elaborated domain GR-estimators are optimal, they have the minimum variance in a class of estimators that satisfy summation restriction. Furthermore, their variances are smaller than the variances of the corresponding initial domain estimators. The variance/covariance formulae of the domain GR-estimators are explicitly given.The ratio estimators as representatives of the non-additive domain estimators are considered. Their design-based covariance matrix, being crucial for the GR-estimator, is presented. Its structure simplifies under certain assumptions on sampling design (and population model). The corresponding simpler forms of the domain GR-estimators are elaborated as well. The hypergeometric [Traat I., Ilves M., 2007. The hypergeometric sampling design, theory and practice. Acta Appl. Math. 97, 311–321] and the simple random sampling designs are considered in more detail. The results are illustrated in a simulation study where the optimal domain estimator displays its superiority among other meaningful domain estimators. It is noteworthy that due to the imposed restrictions also these other estimators, though not optimal, can be much more precise than the initial estimators.     

Finite-population variance estimation under systematic sampling schemes with multiple random starts

Motivated by Sampath [Finite population variance estimation under LSS with multiple random starts, Commun. Statist. – Theory Methods 38 (2009), pp. 3596–3607], in this paper unbiased estimators for population variance have been developed under linear systematic sampling, balanced systematic sampling and modified systematic sampling with multiple random starts. Expressions for variances of the estimators are also developed. Detailed numerical comparative studies have been carried out to study the performances of the estimators under various systematic sampling schemes with multiple random starts and some interesting conclusions have been drawn out of the study.