Fitting Variance Components Model and Fixed Effects Model for One-Way Analysis of Variance to Complex Survey Data

Under complex survey sampling, in particular when selection probabilities depend on the response variable (informative sampling), the sample and population distributions are different, possibly resulting in selection bias. This article is concerned with this problem by fitting two statistical models, namely: the variance components model (a two-stage model) and the fixed effects model (a single-stage model) for one-way analysis of variance, under complex survey design, for example, two-stage sampling, stratification, and unequal probability of selection, etc. Classical theory underlying the use of the two-stage model involves simple random sampling for each of the two stages. In such cases the model in the sample, after sample selection, is the same as model for the population; before sample selection. When the selection probabilities are related to the values of the response variable, standard estimates of the population model parameters may be severely biased, leading possibly to false inference. The idea behind the approach is to extract the model holding for the sample data as a function of the model in the population and of the first order inclusion probabilities. And then fit the sample model, using analysis of variance, maximum likelihood, and pseudo maximum likelihood methods of estimation. The main feature of the proposed techniques is related to their behavior in terms of the informativeness parameter. We also show that the use of the population model that ignores the informative sampling design, yields biased model fitting.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

中国农产量调查中几种可行的PPS系统抽样设计

PPS系统抽样设计是国际上比较流行的设计,它在实践中有着广泛的应用。我国农产量调查采用的是对称等距抽样设计,它实际上可理解为是一种变形的PPS系统抽样设计,使用的是以播种面积作为辅助变量。文章给出了几种可行的PPS系统抽样设计,辅助变量分别取为农户数、播种面积、耕地面积、切块数,并对这几种抽样设计进行了应用分析。     

Design‐Based Inference in a Mixture Model for Ordinal Variables for a Two Stage Stratified Design

In this paper we present methods for inference on data selected by a complex sampling design for a class of statistical models for the analysis of ordinal variables. Specifically, assuming that the sampling scheme is not ignorable, we derive for the class of cub models (Combination of discrete Uniform and shifted Binomial distributions) variance estimates for a complex two stage stratified sample. Both Taylor linearization and repeated replication variance estimators are presented. We also provide design‐based test diagnostics and goodness‐of‐fit measures. We illustrate by means of real data analysis the differences between survey‐weighted and unweighted point estimates and inferences for cub model parameters.     

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.     

An Optimal Procedure for Sample Coordination Using Multiple Objective Functions and Nearest Proportional to IPPS Size Sampling Designs

The sample coordination problem involves maximization or minimization of overlap of sampling units in different/repeated surveys. Several optimal techniques using transportation theory, controlled rounding, and controlled selection have been suggested in literature to solve the sample coordination problem. In this article, using the multiple objective programming, we propose a method for sample coordination which facilitates variance estimation using the Horvitz–Thompson estimator. The proposed procedure can be applied to any two-sample surveys having identical universe and stratification. Some examples are discussed to demonstrate the utility of the proposed procedure.     

缺失偏态数据下联合位置与尺度模型的统计推断

为了研究缺失偏态数据下的联合位置与尺度模型,基于分布自身的特点,提出了一种适合缺失偏态数据下联合建模的插补方法———修正随机回归插补方法,该方法对缺失数据下模型偏度参数的调整十分显著。通过随机模拟和实例研究,并与回归插补和随机回归插补方法进行比较,结果表明,所提出的修正随机回归插补方法是有用和有效的。     

Bayesian experimental design for a toxicokinetic-toxicodynamic model

The aim of this study is to apply the Bayesian method of identifying optimal experimental designs to a toxicokinetic-toxicodynamic model that describes the response of aquatic organisms to time dependent concentrations of toxicants. As for experimental designs, we restrict ourselves to pulses and constant concentrations. A design of an experiment is called optimal within this set of designs if it maximizes the expected gain of knowledge about the parameters. Focus is on parameters that are associated with the auxiliary damage variable of the model that can only be inferred indirectly from survival time series data. Gain of knowledge through an experiment is quantified both with the ratio of posterior to prior variances of individual parameters and with the entropy of the posterior distribution relative to the prior on the whole parameter space. The numerical methods developed to calculate expected gain of knowledge are expected to be useful beyond this case study, in particular for multinomially distributed data such as survival time series data.     

Nonparametric Estimation for Random Censored Data Based on Ranking Set Sampling

In the case where the population distribution is unknown, the Kaplan–Meier estimator of the reliability function based on a ranked set sample with random right-censored data is first proposed. It is shown to be a unique self-consistent estimator. Then, the censored RSS estimator of the population mean is constructed. A simulation study is conducted to compare the performance of the proposed estimators with the corresponding estimators based on a simple random sample. It is shown that the ranked set sampling has higher efficiency. Finally, the proposed method is applied to a renal carcinoma study.     

Adaptive LASSO for linear mixed model selection via profile log-likelihood

Mixed model selection is quite important in statistical literature. To assist the mixed model selection, we employ the adaptive LASSO penalized term to propose a two-stage selection procedure for the purpose of choosing both the random and fixed effects. In the first stage, we utilize the penalized restricted profile log-likelihood to choose the random effects; in the second stage, after the random effects are determined, we apply the penalized profile log-likelihood to select the fixed effects. In each stage, the Newton–Raphson algorithm is performed to complete the parameter estimation. We prove that the proposed procedure is consistent and possesses the oracle properties. The simulations and a real data application are conducted for demonstrating the effectiveness of the proposed selection procedure.     

Reference priors for shrinkage and smoothing parameters

A reference prior and corresponding reference posteriors are derived for a basic Normal variance components model with two components. Different parameterizations are considered, in particular one in terms of a shrinkage or smoothing parameter. Earlier results for the one-way ANOVA setting are generalized and a broad range of applications of the general results is indicated. Numerical examples of application to spline smoothing are given for illustration and the results compared with other well-known techniques considered to be “non-informative” about the smoothing parameter.     

Interval estimators for reliability: the bivariate normal case

This paper proposes procedures to provide confidence intervals (CIs) for reliability in stress–strength models, considering the particular case of a bivariate normal set-up. The suggested CIs are obtained by employing either asymptotic variances of maximum-likelihood estimators or a bootstrap procedure. The coverage and the accuracy of these intervals are empirically checked through a simulation study and compared with those of another proposal in the literature. An application to real data is provided.     

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.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.