Model‐Based Non‐parametric Variance Estimation for Systematic Sampling

Abstract. Systematic sampling is frequently used in surveys, because of its ease of implementation and its design efficiency. An important drawback of systematic sampling, however, is that no direct estimator of the design variance is available. We describe a new estimator of the model‐based expectation of the design variance, under a non‐parametric model for the population. The non‐parametric model is sufficiently flexible that it can be expected to hold at least approximately in many situations with continuous auxiliary variables observed at the population level. We prove the model consistency of the estimator for both the anticipated variance and the design variance under a non‐parametric model with a univariate covariate. The broad applicability of the approach is demonstrated on a dataset from a forestry survey.     

Inference on Polychotomous Responses in Finite Populations

Abstract. A model‐based predictive estimator is proposed for the population proportions of a polychotomous response variable, based on a sample from the population and on auxiliary variables, whose values are known for the entire population. The responses for the non‐sample units are predicted using a multinomial logit model, which is a parametric function of the auxiliary variables. A bootstrap estimator is proposed for the variance of the predictive estimator, its consistency is proved and its small sample performance is compared with that of an analytical estimator. The proposed predictive estimator is compared with other available estimators, including model‐assisted ones, both in a simulation study involving different sampling designs and model mis‐specification, and using real data from an opinion survey. The results indicate that the prediction approach appears to use auxiliary information more efficiently than the model‐assisted approach.     

Inference for domains under imputation for missing survey data

The authors study the estimation of domain totals and means under survey‐weighted regression imputation for missing items. They use two different approaches to inference: (i) design‐based with uniform response within classes; (ii) model‐assisted with ignorable response and an imputation model. They show that the imputed domain estimators are biased under (i) but approximately unbiased under (ii). They obtain a bias‐adjusted estimator that is approximately unbiased under (i) or (ii). They also derive linearization variance estimators. They report the results of a simulation study on the bias ratio and efficiency of alternative estimators, including a complete case estimator that requires the knowledge of response indicators.     

Variance estimation when donor imputation is used to fill in missing values

Donor imputation is frequently used in surveys. However, very few variance estimation methods that take into account donor imputation have been developed in the literature. This is particularly true for surveys with high sampling fractions using nearest donor imputation, often called nearest‐neighbour imputation. In this paper, the authors develop a variance estimator for donor imputation based on the assumption that the imputed estimator of a domain total is approximately unbiased under an imputation model; that is, a model for the variable requiring imputation. Their variance estimator is valid, irrespective of the magnitude of the sampling fractions and the complexity of the donor imputation method, provided that the imputation model mean and variance are accurately estimated. They evaluate its performance in a simulation study and show that nonparametric estimation of the model mean and variance via smoothing splines brings robustness with respect to imputation model misspecifications. They also apply their variance estimator to real survey data when nearest‐neighbour imputation has been used to fill in the missing values. The Canadian Journal of Statistics 37: 400–416; 2009 © 2009 Statistical Society of Canada     

Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

A NOTE ON VARIANCE ESTIMATION FOR THE GENERALIZED REGRESSION PREDICTOR

The generalized regression (GREG) predictor is used for estimating a finite population total when the study variable is well‐related to the auxiliary variable. In 1997, Chaudhuri & Roy provided an optimal estimator for the variance of the GREG predictor within a class of non‐homogeneous quadratic estimators (H) under a certain superpopulation model M. They also found an inequality concerning the expected variances of the estimators of the variance of the GREG predictor belonging to the class H under the model M. This paper shows that the derivation of the optimal estimator and relevant inequality, presented by Chaudhuri & Roy, are incorrect.     

人口数目估计中的独立双系统估计量研究

利用存在统计相依关系的两份人口登记名单构造的非独立双系统估计量是目前估计总体实际人口数的前沿方法。该估计量由最初用于估计一个区域内的野生动物数目的捕获-再捕获模型移植而来。非独立双系统估计量的一个明显缺陷是低估总体实际人口数。用独立双系统估计量替代非独立双系统估计量属于人口数目估计领域的理论创新研究。采用数理分析与实证分析相结合的方法研究独立双系统估计量及其方差估计量。为便于读者理解,通过一个实证案例全面演示了独立双系统估计量的计算过程。研究表明,独立双系统估计量所估计的人口数平均接近于实际人口数,建议在未来人口数目估计中应用独立双系统估计量。     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

Series Estimation in Partially Linear In‐Slide Regression Models

Abstract. The partially linear in‐slide model (PLIM) is a useful tool to make econometric analyses and to normalize microarray data. In this article, by using series approximations and a least squares procedure, we propose a semiparametric least squares estimator (SLSE) for the parametric component and a series estimator for the non‐parametric component. Under weaker conditions than those imposed in the literature, we show that the SLSE is asymptotically normal and that the series estimator attains the optimal convergence rate of non‐parametric regression. We also investigate the estimating problem of the error variance. In addition, we propose a wild block bootstrap‐based test for the form of the non‐parametric component. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on a set of economical data is also illustrated.     

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.     

Doubly Robust Inference for the Distribution Function in the Presence of Missing Survey Data

Item non‐response in surveys occurs when some, but not all, variables are missing. Unadjusted estimators tend to exhibit some bias, called the non‐response bias, if the respondents differ from the non‐respondents with respect to the study variables. In this paper, we focus on item non‐response, which is usually treated by some form of single imputation. We examine the properties of doubly robust imputation procedures, which are those that lead to an estimator that remains consistent if either the outcome variable or the non‐response mechanism is adequately modelled. We establish the double robustness property of the imputed estimator of the finite population distribution function under random hot‐deck imputation within classes. We also discuss the links between our approach and that of Chambers and Dunstan. The results of a simulation study support our findings.     

Variance estimation for the finite population distribution function with complete auxiliary information

The authors develop jackknife and analytical variance estimators for the estimator of Chambers & Dunstan (1986) and Rao, Kovar & Mantel (1990) of the finite population distribution function, using complete auxiliary information. They also describe the associated model and show the design consistency of the variance estimators, whose small‐sample performance is examined through a limited simulation study. They highlight the operational advantages of the jackknife in the model‐based setting of Chambers & Dunstan (1986) and its better conditional performance in the design‐based setting of Rao, Kovar & Mantel (1990).     

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.     

Shape Constrained Non‐parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model

Abstract. We investigate non‐parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non‐decreasing baseline hazard function are proposed. We derive the non‐parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left‐hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non‐increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non‐increasing baseline density, defined as the left‐hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

Estimation of the mean of a spherically symmetric distribution with constraints on the norm

The authors consider the problem of estimating, under quadratic loss, the mean of a spherically symmetric distribution when its norm is supposed to be known and when a residual vector is available. They give a necessary and sufficient condition for the optimal James‐Stein estimator to dominate the usual estimator. Various examples are given that are not necessarily variance mixtures of normal distributions. Consideration is also given to an alternative class of robust James‐Stein type estimators that take into account the residual vector. A more general domination condition is given for this class.     

Efficient Estimation of an Additive Quantile Regression Model

Abstract. In this paper, two non‐parametric estimators are proposed for estimating the components of an additive quantile regression model. The first estimator is a computationally convenient approach which can be viewed as a more viable alternative to existing kernel‐based approaches. The second estimator involves sequential fitting by univariate local polynomial quantile regressions for each additive component with the other additive components replaced by the corresponding estimates from the first estimator. The purpose of the extra local averaging is to reduce the variance of the first estimator. We show that the second estimator achieves oracle efficiency in the sense that each estimated additive component has the same variance as in the case when all other additive components were known. Asymptotic properties are derived for both estimators under dependent processes that are strictly stationary and absolutely regular. We also provide a demonstrative empirical application of additive quantile models to ambulance travel times.     

Partially Calibrated Poststratified Weights for Handling Unit Non Response

We consider a non response-adjusted poststratified estimation when there exists a set of clear response homogeneity groups but the population distribution of that set is unknown, which is common in practice. We propose a partially calibrated poststratified estimator that is asymptotically unbiased and satisfies a calibration equation for the auxiliary variables of which the joint population distribution is known. We also provide a variance estimator of the proposed poststratified estimator. In a small simulation study, the proposed estimator performed better than or comparable to commonly used estimators.     

On the Estimation of the Density of a Directional Data Stream

Many directional data such as wind directions can be collected extremely easily so that experiments typically yield a huge number of data points that are sequentially collected. To deal with such big data, the traditional nonparametric techniques rapidly require a lot of time to be computed and therefore become useless in practice if real time or online forecasts are expected. In this paper, we propose a recursive kernel density estimator for directional data which (i) can be updated extremely easily when a new set of observations is available and (ii) keeps asymptotically the nice features of the traditional kernel density estimator. Our methodology is based on Robbins–Monro stochastic approximations ideas. We show that our estimator outperforms the traditional techniques in terms of computational time while being extremely competitive in terms of efficiency with respect to its competitors in the sequential context considered here. We obtain expressions for its asymptotic bias and variance together with an almost sure convergence rate and an asymptotic normality result. Our technique is illustrated on a wind dataset collected in Spain. A Monte‐Carlo study confirms the nice properties of our recursive estimator with respect to its non‐recursive counterpart.