Multiple-start balanced modified systematic sampling in the presence of linear trend

ABSTRACT

In this paper, we propose a sampling design termed as multiple-start balanced modified systematic sampling (MBMSS), which involves the supplementation of two or more balanced modified systematic samples, thus permitting us to obtain an unbiased estimate of the associated sampling variance. There are five cases for this design and in the presence of linear trend only one of these cases is optimal. To further improve results for the other cases, we propose an estimator that removes linear trend by applying weights to the first and last sampling units of the selected balanced modified systematic samples and is thus termed as the MBMSS with end corrections (MBMSSEC) estimator. By assuming a linear trend model averaged over a super-population model, we will compare the expected mean square errors (MSEs) of the proposed sample means, to that of simple random sampling (SRS), linear systematic sampling (LSS), stratified random sampling (STR), multiple-start linear systematic sampling (MLSS), and other modified MLSS estimators. As a result, MBMSS is optimal for one of the five possible cases, while the MBMSSEC estimator is preferred for three of the other four cases.     

Circular versions of systematic sampling in the presence of linear trend

In this paper, an alternative approach was employed and derived the explicit expressions of sample mean and its variance for the circular and the diagonal circular systematic schemes since these expressions were unavailable in the existing literature. The study also found the corrected estimator of population mean and average variance under super-population model for these scheme. It further revealed that the circular systematic sampling almost performed better than that of the diagonal circular systematic sampling in both linear and auto-correlated super-population models.     

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.     

Some estimation procedures using a linear model in successive sampling

ABSTRACT

In successive sampling some recent works depict the use of super-population models where information on stable auxiliary variable over occasions has been utilized. Stability character of auxiliary variable may not sustain, if the duration between occasions is large. To cope with such situations, the present work is an attempt to develop some estimation procedures by utilizing the information on two independent auxiliary variables through a linear super-population model. Some estimators are proposed to estimate the current population mean in two occasions successive (rotation) sampling. Optimum replacement strategies are formulated and performances of the proposed estimators have been discussed. Results are interpreted through empirical studies.     

Harmonic mean approach to unbalanced mixed effects models under heteroscedastic variances

In this paper we consider unbalanced mixed models (Scheffe's model) under heteroscedastic variances. By using the harmonic mean approach, It is shown that the problems appear to be anologous to those problems from balanced mixed models under homoscedastic variance. Thus, by using harmonic mean approach, statistical inferences about fixed effects and variance components are derived by using those from balanced models under homoscedastic variance. Laguerre polynomial expansion is used Lo approximate sampling distributions of relevant statistics.     

On the performance of the nearest proportional to size sampling design

Assuming a super-population model the expected variance of the generalized difference estimator (Basu,1971) based on the nearest proportional to size sampling design introduced by Gabler(1987) is shown to be less than that of the same estimator based on an arbitrary sampling design from which the former design is realized. The former strategy is also shown to fare better than an unbiased ratio-cum-generalized difference estimator based on the nearest proportional to size sampling design in the sense of having less expected design variance under the same model.     

HORVITZ-THOMPSON AND DES RAJ ESTIMATORS REVISITED

In this paper it is shown that the generalized πPS sampling strategy consisting of the design with π_i, the probability of inclusion of the ith unit in the sample, proportional to the modified size together with the corresponding Horvitz-Thompson estimator (Rao, 1971), is superior to the symmetrized Des Raj strategy under a general super-population set-up for all values of the super-population parameter g, when the samples are of size two.     

A NOTE ON COMPETING VARIANCE ESTIMATORS IN RANDOMISED RESPONSE SURVEYS

When gathering randomised rather than direct responses on a variable of interest relating to sensitive issues, one may use a modified version of the well-known generalised regression predictor of a finite population total. To construct confidence intervals, this paper proposes four alternative variance estimators – modifications to those usable with direct responses – and examines their relative efficiencies through simulations from simple super-population models.     

Harmonic mean approach to uunbalanced random effects models under heteroscedasticity

In this paper we consider unbalanced random effects models under heteroscedastic variances. By using' the harmonic mean approach, it is shown that the problems are analogous to those from balanced random effects models under horaoscedastic variances. Thus, by using the harmonic mean approach, statistical inferences about variance components are derived by using procedures from balanced models under homoscedastic variances. Laguerre polynomial expansion is used to approximate the sampling distributions of relevant statistics.     

Asymptotic design-cum-model based estimation of variances of estimated linear regression coefficients in survey sampling with unequal probabilities

Postulating a super-population linear regression model for a variable of interest on an auxiliary variable we consider design-based estimation of regression coefficients on drawing a sample with unequal probabilities from a survey population. Asymptotic design-cum-model based variance estimation procedures are proposed.     

Admissible and optimal sampling strategy for estimating finite population mean in randomized response surveys with multiple responses

ABSTRACT

We consider the problem of estimation of a finite population mean (or proportion) related to a sensitive character under a randomized response model when independent responses are obtained from each sampled individual as many times as he/she is selected in the sample and prove the admissibility of a sampling strategy in a class of comparable linear unbiased strategies. We prove that the admissible strategy is also optimal in this class under a super-population model.     

A comparison of alternative variance estimation strategies for estimating the slope of a linear regression in sample surveys

The balanced half-sample, jackknife and linearization methods are used to estimate the variance of the slope of a linear regression under a variety of computer generated situations. The basic sampling design is one in which two PSU's are selected from each of a number of strata . The variance estimation techniques are compared with a Monte Carlo experiment. Results show that variance estimates may be highly biased and variable unless sizeable numbers of observations are available from each stratum. The jackknife and linearization estimates appear superior to the balanced half sample method - particularly when the number of strata or the number of available observations from each stratum is small.     

Estimating the variances of ratio estimates in complex sample surveys with two primary sampling units per stratum—a comparison of balanced replication and jackknife techniques

The balanced half-sample and jackknife variance estimation techniques are used to estimate the variance of the combined ratio estimate. An empirical sampling study is conducted using computer-generated populations to investigate the variance, bias and mean square error of these variance estimators and results are compared to theoretical results derived elsewhere for the linear case. Results indicate that either the balanced half-sample or jackknife method may be used effectively for estimating the variance of the combined ratio estimate.     

基于超总体模型的设计效应分解

本文基于超总体模型研究抽样调查中设计效应的计算问题。首先以随机效应模型为基础，明确了简单随机、二阶段、不等概率和分层抽样对应的超总体模型，进而通过所给模型推导出分层、类集、加权单因素设计效应的计算公式和多因素组合的设计效应计算公式并给出了对应估计量，公式表明：多因素同时存在的组合设计效应等于对应单因素设计效应的乘积。最后，对设计效应的理论值、估计值和真实值之间的关系进行了蒙特卡洛仿真，并利用相对偏倚、相对均方误进行了评价。本文的研究，对复杂抽样设计中正确计算、使用设计效应具有指导意义。     

Finite Population Variance Estimation Under LSS with Multiple Random Starts

In this article, an unbiased estimator for finite population variance is developed under linear systematic sampling with two random starts and an explicit expression for its variance is also obtained. The study is supported by two real life situations. A detailed numerical comparative study has been carried out to compare its average variance with the average variance of the conventional unbiased estimator for finite population variance under simple random sampling for a wide variety of populations. Results based on the study strongly favor the use of the developed estimator for such populations.     

Remainder modified systematic sampling in the presence of linear trend

If the population size is not a multiple of the sample size, then the usual linear systematic sampling design is unattractive, since the sample size obtained will either vary, or be constant and different to the required sample size. Only a few modified systematic sampling designs are known to deal with this problem and in the presence of linear trend, most of these designs do not provide favorable results. In this paper, a modified systematic sampling design, known as remainder modified systematic sampling (RMSS), is introduced. There are seven cases of RMSS and the results in this paper suggest that the proposed design is favorable, regardless of each case, while providing linear trend-free sampling results for three of the seven cases. To obtain linear trend-free sampling for the other cases and thus improve results, an end corrections estimator is constructed.     

Prediction in a class of mixed models with two variance components

A best unbiased predictor (BUP) of an arbitrary linear combination of fixed and random effects in mixed linear models is available when the true values of the variance ratios are known. When the true values are unknown, a two-stage predictor, obtained from the BUP by replacing the true values by estimated values, can be used. In this article, exact mean squared errors of two-stage predictors are obtained for a class of mixed models with two variance components that includes the balanced one-way random model and other analysis-of-variance models with proportional frequencies and one balanced random factor.     

A Further Study on Reliable Life Estimation under Ranked Set Sampling

This article considers nonparametric estimation of reliable life based on ranked set sampling and its properties. It is proven analytically that the large sample efficiency of the reliable life estimator under the balanced ranked set sampling is higher than that under the simple random sampling of the same size, but the relative efficiency damps away as the reliable life moves away from the median on both directions. To improve the efficiency for the estimation of extreme reliable life, we then propose a reliable life estimator under a modified ranked set sampling protocol, its strong consistency and asymptotic normality are established. The proposed sampling is shown to be superior to the balanced ranked set sampling, and the relative advantage improves as the reliable life moves away from median. Finally, results of simulation studies for small sample as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

A Comparison of Estimation Methods on the Coverage Probability of Satterthwaite Confidence Intervals for Assay Precision with Unbalanced Data

Construction of closed-form confidence intervals on linear combinations of variance components were developed generically for balanced data and studied mainly for one-way and two-way random effects analysis of variance models. The Satterthwaite approach is easily generalized to unbalanced data and modified to increase its coverage probability. They are applied on measures of assay precision in combination with (restricted) maximum likelihood and Henderson III Type 1 and 3 estimation. Simulations of interlaboratory studies with unbalanced data and with small sample sizes do not show superiority of any of the possible combinations of estimation methods and Satterthwaite approaches on three measures of assay precision. However, the modified Satterthwaite approach with Henderson III Type 3 estimation is often preferred above the other combinations.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.