Variance Of Proposed Estimator Research Articles

Bootstrap Variance Estimation for Semiparametric Finite Population Distribution Function Estimator

Estimating finite population distribution function (FPDF) emerges as an important problem to the survey statisticians since the pioneering work of Chambers and Dunstan [1] . It unifies estimation of standard finite population parameters, namely, mean and quantiles. Regarding this, estimating variance of FPDF estimator is an important task for accessing the quality of the estimtor and drawing inferences (e.g., confidence interval estimation) on finite population parameters. Due to non-linearity of FPDF estimator, resampling-based methods are developed earlier for parametric or non-parametric Chambers–Dunstan estimator. Here, we attempt the problem of estimating variance of P-splines-based semiparametric model-based Chambers–Dunstan type estimator of the FPDF. The proposed variance estimator involes bootstrapping. Here, the bootstrap procedure is non-trivial since it does not imitate the full mechanism of two-stage sample generating procedure from an infinite hypothetical population (superpopulation). We have established the weak consistency of the proposed resampling-based variance estimator for specific sampling designs, e.g., simple random sampling. Also, the satisfactory empirical performance of the poposed estimator has been shown through simulation studies and a real life example.

Generalized variance estimators in adaptive cluster sampling using single auxiliary variable

This study presents generalized variance estimators utilizing the information on single auxiliary variable in adaptive cluster sampling for finite population variance. The expressions for mean square errors and biases for proposed variance estimators are obtained. The result of simulation study presents that the suggested variance estimators for adaptive cluster sampling (ACS) design are most efficient for population variance as compare to Isaki (1983) variance estimator for SRSWOR.

Extrapolation techniques in U-statistic variance estimation

ABSTRACTThis article considers the problem of variance estimation of a U-statistic. Following the proposal of a linearly extrapolated variance estimator in Wang and Chen (2015), we consider a second-order extrapolation technique and devise a variance estimator that is nearly second-order unbiased. Simulation studies confirm that the second-order extrapolated variance estimator has smaller bias than the linearly extrapolated variance estimator and the jackknife variance estimator across a wide selection of distributions. In addition, the proposal also yields a smaller mean squared error than its counterparts. In the end, we discuss the advantages of the proposed variance estimator in regression analysis and model selection.

Testing discontinuities in nonparametric regression

ABSTRACTIn nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13] and propose to further improve it. To achieve the goal, we first reveal that their method is less efficient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the residual variance and the total amount of discontinuities simultaneously. In both theory and simulation, we show that the proposed variance estimator has a smaller mean-squared error compared to the existing estimator, whereas the estimation efficiency for the total amount of discontinuities remains unchanged. Finally, we construct a new test procedure for detection of discontinuities using the proposed method; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings.

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

ABSTRACTWe 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 model regression estimation of a spatial total in the continuous plane paradigm

In environmental studies, it is a common practice to follow the continuous-plane paradigm for estimating parameters of interest, like totals or means, of spatial variables defined over a surface whose points correspond to points of a connected and compact subset of R2. In the design-based approach, randomness comes from the random sampling of points in the domain, wherein to observe the values of the spatial variable. Adopting this type of approach, under a model-assisted framework, in this paper we present a class of estimators of the total of the spatial variable, which are based on linear mixed models. We introduce first the mixed model regression estimator of the total and its properties. Methods for choosing the smoothing parameter are also discussed. Then, we present the penalized splines regression estimator of the total, as a member of the class, and its asymptotic properties when the number of knots increases with the sample size within an infill asymptotic framework. This estimator, under mild conditions, is proved to be asymptotically unbiased, consistent, asymptotically normally distributed, and more efficient than the basic unbiased estimator. Furthermore, it can be super-efficient, that is, the variance order may be higher than the usual n−1, according to the degree of smoothness of the spatial variable. A simulation study shows the performance of the estimator as well as that of the proposed variance estimators.

On variance estimate for covariate adjustment by propensity score analysis.

Propensity score (PS) methods have been used extensively to adjust for confounding factors in the statistical analysis of observational data in comparative effectiveness research. There are four major PS-based adjustment approaches: PS matching, PS stratification, covariate adjustment by PS, and PS-based inverse probability weighting. Though covariate adjustment by PS is one of the most frequently used PS-based methods in clinical research, the conventional variance estimation of the treatment effects estimate under covariate adjustment by PS is biased. As Stampf et al. have shown, this bias in variance estimation is likely to lead to invalid statistical inference and could result in erroneous public health conclusions (e.g., food and drug safety and adverse events surveillance). To address this issue, we propose a two-stage analytic procedure to develop a valid variance estimator for the covariate adjustment by PS analysis strategy. We also carry out a simple empirical bootstrap resampling scheme. Both proposed procedures are implemented in an R function for public use. Extensive simulation results demonstrate the bias in the conventional variance estimator and show that both proposed variance estimators offer valid estimates for the true variance, and they are robust to complex confounding structures. The proposed methods are illustrated for a post-surgery pain study. Copyright © 2016 John Wiley & Sons, Ltd.

잔차 수정을 이용한 불연속 분산함수의 비모수적 추정

In usual, the discontinuous variance function was estimated nonparametrically using a kernel type estimator with data sets split by an estimated location of the change point. Kang et al. (2000) proposed the Gasser- type kernel estimator of the discontinuous regression function using the adjusted observations of response variable by the estimated jump size of the change point in (1992). The adjusted observations might be a random sample coming from a continuous regression function. In this paper, we estimate the variance function using the Nadaraya-Watson kernel type estimator using the adjusted squared residuals by the estimated location of the change point in the discontinuous variance function like Kang et al. (2000) did. The rate of convergence of integrated squared error of the proposed variance estimator is derived and numerical work demonstrates the improved performance of the method over the exist one with simulated examples.

Easy and accurate variance estimation of the nonparametric estimator of the partial area under the ROC curve and its application.

The receiver operating characteristic (ROC) curve is a popular technique with applications, for example, investigating an accuracy of a biomarker to delineate between disease and non-disease groups. A common measure of accuracy of a given diagnostic marker is the area under the ROC curve (AUC). In contrast with the AUC, the partial area under the ROC curve (pAUC) looks into the area with certain specificities (i.e., true negative rate) only, and it can be often clinically more relevant than examining the entire ROC curve. The pAUC is commonly estimated based on a U-statistic with the plug-in sample quantile, making the estimator a non-traditional U-statistic. In this article, we propose an accurate and easy method to obtain the variance of the nonparametric pAUC estimator. The proposed method is easy to implement for both one biomarker test and the comparison of two correlated biomarkers because it simply adapts the existing variance estimator of U-statistics. In this article, we show accuracy and other advantages of the proposed variance estimation method by broadly comparing it with previously existing methods. Further, we develop an empirical likelihood inference method based on the proposed variance estimator through a simple implementation. In an application, we demonstrate that, depending on the inferences by either the AUC or pAUC, we can make a different decision on a prognostic ability of a same set of biomarkers. Copyright © 2016 John Wiley & Sons, Ltd.

Precision of systematic and random sampling in clustered populations: habitat patches and aggregating organisms.

Natural populations of plants and animals spatially cluster because (1) suitable habitat is patchy, and (2) within suitable habitat, individuals aggregate further into clusters of higher density. We compare the precision of random and systematic field sampling survey designs under these two processes of species clustering. Second, we evaluate the performance of 13 estimators for the variance of the sample mean from a systematic survey. Replicated simulated surveys, as counts from 100 transects, allocated either randomly or systematically within the study region, were used to estimate population density in six spatial point populations including habitat patches and Matérn circular clustered aggregations of organisms, together and in combination. The standard one-start aligned systematic survey design, a uniform 10 x 10 grid of transects, was much more precise. Variances of the 10 000 replicated systematic survey mean densities were one-third to one-fifth of those from randomly allocated transects, implying transect sample sizes giving equivalent precision by random survey would need to be three to five times larger. Organisms being restricted to patches of habitat was alone sufficient to yield this precision advantage for the systematic design. But this improved precision for systematic sampling in clustered populations is underestimated by standard variance estimators used to compute confidence intervals. True variance for the survey sample mean was computed from the variance of 10 000 simulated survey mean estimates. Testing 10 published and three newly proposed variance estimators, the two variance estimators (v) that corrected for inter-transect correlation (ν₈ and ν(W)) were the most accurate and also the most precise in clustered populations. These greatly outperformed the two "post-stratification" variance estimators (ν₂ and ν₃) that are now more commonly applied in systematic surveys. Similar variance estimator performance rankings were found with a second differently generated set of spatial point populations, ν₈ and ν(W) again being the best performers in the longer-range autocorrelated populations. However, no systematic variance estimators tested were free from bias. On balance, systematic designs bring more narrow confidence intervals in clustered populations, while random designs permit unbiased estimates of (often wider) confidence interval. The search continues for better estimators of sampling variance for the systematic survey mean.

Estimating the variance of decomposition effects

ABSTRACTWe derive the asymptotic variance of the Blinder–Oaxaca decomposition effects. We show that the delta method approach that builds on the assumption of fixed regressors understates true variability of the decomposition effects when regressors are stochastic. Our proposed variance estimator takes randomness of regressors into consideration. Our approach is applicable to both the linear and nonlinear decompositions. Previously, only a bootstrap method has been a valid option for nonlinear decompositions. As our derivation follows the general framework of m-estimation, it is straightforward to extend our variance estimator to a cluster-robust variance estimator. We demonstrate the finite-sample performance of our variance estimator with a Monte Carlo study and present a real-data application.

A general class of linearly extrapolated variance estimators

A general class of linearly extrapolated variance estimators was developed as an extension of the conventional leave-one-out jackknife variance estimator. In the context of U-statistic variance estimation, the proposed variance estimator is first-order unbiased. After showing the equivalence between the Hoeffding decomposition (Hoeffding, 1948) and the ANOVA decomposition (Efron and Stein, 1981), we study the bias property of the proposed variance estimator in comparison to the conventional jackknife method. Simulation studies indicate that the proposal has comparable performance to the jackknife method when assessing the variance of the sample variance in various distributions. An application to half-sampling cross-validation indicates that the proposal is more computationally efficient and shows better performance than its jackknife counterpart in the context of regression analysis.

Estimation of Population Variance from Multi-ranker Ranked Set Sampling Designs

In this article, we develop an estimator for a population variance based on a multi-ranker ranked set sampling design. In a multi-ranker design, the units are ranked by more than one ranker allowing ties whenever the confidence level of the rankers is low. The ranking information of all rankers is then combined in a meaningful way to create a single measure. This measure is used to construct the sampling design and a new estimator for the population variance. The article investigates the bias and relative efficiency of the proposed variance estimator. It is shown that the new estimator performs as good as or better than its competitors in the literature.

Variance Estimation from Complex Survey Designs: A Case Study of Household Income and Expenditure Survey Design 2002/03, Botswana

In large-scale surveys, it is common to use a multistage sampling design, where the first-stage units are selected with varying probability without replacement and the second-stage units are selected according to the systematic sampling procedure. It is however well known that this sampling design makes it impossible to find unbiased estimate of the variance of the estimate of the population total. In this paper, a few methods of variance estimation in multistage sampling involving systematic sampling design have been proposed to overcome the problem discussed. The proposed variance estimators are compared with the few conventional methods, for example, Random group, Jackknife, as well as the method used by the Central Statistical Office (CSO) using the live data based on Household Income and Expenditure Survey (HIES) 2002/03, Botswana, collected by CSO. It is found that all the proposed methods and Jackknife method perform better than the method used by CSO.

Weighted kappa statistic for clustered matched-pair ordinal data

As an important extension of the regular kappa statistic, the weighted kappa statistic has been widely used to assess the agreement between two procedures for independent matched-pair ordinal data. For clustered matched-pair ordinal data, based on the delta method and sampling techniques, a non-parametric variance estimator for the weighted kappa statistic is proposed without within-cluster correlation structure or distributional assumptions. The results of an extensive Monte Carlo simulation study demonstrate that the proposed weighted kappa statistic provides consistent estimation, and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters (e.g., K≥50). Compared with the variance estimator ignoring dependence within a cluster, the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra-cluster correlation is fair (ρ≥0.3), with more pronounced improvement when ρ is further increased. Moreover, under the general analysis of variance setting with systematic variability between procedures and clusters being included as a component of total variation, the equivalence between weighted kappa statistic and intraclass correlation coefficient is established. To illustrate the practical application of the proposed estimator, two real medical research data examples of clustered matched-pair ordinal data are analyzed, including an agreement study to compare two methods for assessing cervical ectopy, and a physician–patients data example from the Enhancing Communication and HIV Outcomes study.

Kappa statistic for clustered matched-pair data

Kappa statistic is widely used to assess the agreement between two procedures in the independent matched-pair data. For matched-pair data collected in clusters, on the basis of the delta method and sampling techniques, we propose a nonparametric variance estimator for the kappa statistic without within-cluster correlation structure or distributional assumptions. The results of an extensive Monte Carlo simulation study demonstrate that the proposed kappa statistic provides consistent estimation and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters (e.g., K ≥50). Compared with the variance estimator ignoring dependence within a cluster, the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra-cluster correlation is fair (ρ ≥0.3), with more pronounced improvement when ρ is further increased. To illustrate the practical application of the proposed estimator, we analyze two real data examples of clustered matched-pair data.

Variance estimation of a general u-statistic with appllication to cross-validation

This paper addresses the problem of variance estimation for a general U-statistic. U-statistics form a class of unbiased estimators for those parameters of interest that can be written as Ef(X 1;:::;Xk)g, where is a symmetric kernel function with k arguments. Although estimating the variance of a U-statistic is clearly of interest, asymptotic results for a general U-statistic are not necessarily reliable when the kernel size k is not negligible compared with the sample size n. Such situations arise in cross-validation and other non- parametric risk estimation problems. On the other hand, the exact closed form variance is complicated in form, especially when both k and n are large. We have devised an unbiased variance estimator for a general U-statistic. It can be written as a quadratic form of the kernel function and is applicable as long as k n=2. In addition, it can be represented in a fa- miliar analysis of variance form as a contrast of between-class and within-class variation. As a further step to make the proposed variance estimator more practical, we developed a parti- tion resampling scheme that can be used to realize the U-statistic and its variance estimator simultaneously with high computational efficiency. A data example in the context of model selection is provided. To study our estimator, we construct a U-statistic cross-validation tool, akin to the BIC criterion for model selection. With our variance estimator we can test which model has the smallest risk.

Variance Estimation and Asymptotic Confidence Bands for the Mean Estimator of Sampled Functional Data with High Entropy Unequal Probability Sampling Designs

ABSTRACTFor fixed size sampling designs with high entropy, it is well known that the variance of the Horvitz–Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional, and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz–Thompson estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of 1 week.

A new replicate variance estimator for unequal probability sampling without replacement

AbstractWe propose a new replicate variance estimator suitable for differentiable functions of estimated totals. The proposed variance estimator is defined for any unequal‐probability without‐replacement sampling design, it naturally includes finite population corrections and it allows two‐stage sampling. We show its design‐consistency and its close relationship with linearization variance estimators. When estimating a total, the proposed estimator reduces to the Horvitz–Thompson variance estimator. Simulations suggest that the proposed variance estimator is more stable than its replicate competitors. The Canadian Journal of Statistics 41: 508–524; 2013 © 2013 Statistical Society of Canada

Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.