The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables

In sample surveys, it is usual to make use of auxiliary information to increase the precision of the estimators. We propose a new chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables and obtain the mean squared error (MSE) equations for the proposed estimators. We find theoretical conditions that make proposed estimators more efficient than the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

Ratio estimation with measurement error in the auxiliary variate

Summary .  With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

An Alternative Approach to Estimation in Two-Phase Sampling Using two Auxiliary Variables

Using two-phase sampling mechanism, two alternative estimators in the presence of the available knowledge on second auxiliary variable z are considered, when the population mean of the main auxiliary variable × is unknown. The suggested estimators are found to be more eficient than the ratio-type and regression-type estimators suggested by KIREGYERA (1980, 1984).     

On the inadmissibility of Watterson's estimator

We consider the estimation of the scaled mutation parameter θ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.     

An Improved Class of Estimators for Finite Population Mean in Sample Surveys

This paper proposes a class of estimators for estimating the finite population mean -Y of a study variate y using information on two auxiliary variates, one of which is positively and the other negatively correlated with the study variate y. An “asymptotically optimum estimator” (AOE) in the class is identified with its bias and mean square error formulae. It is observed that the proposed AOE is more efficient than Srivastava (1965), Srivastava (1974), Prasad (1989) and Gandge , Varghese , and Prabhu-Ajgaonkar (1993) estimators.     

The dilemma of negative analysis of variance estimators of intraclass correlation

Summary At least two common practices exist when a negative variance component estimate is obtained, either setting it to zero or not reporting the estimate. The consequences of these practices are investigated in the context of the intraclass correlation estimation in terms of bias, variance and mean squared error (MSE). For the one-way analysis of variance random effects model and its extension to the common correlation model, we compare five estimators: analysis of variance (ANOVA), concentrated ANOVA, truncated ANOVA and two maximum likelihood-like (ML) estimators. For the balanced case, the exact bias and MSE are calculated via numerical integration of the exact sample distributions, while a Monte Carlo simulation study is conducted for the unbalanced case. The results indicate that the ANOVA estimator performs well except for designs with family size n = 2. The two ML estimators are generally poor, and the concentrated and truncated ANOVA estimators have some advantages over the ANOVA in terms of MSE. However, the large biases may make the concentrated and truncated ANOVA estimators objectionable when intraclass correlation () is small. Bias should be a concern when a pooled estimate is obtained from the literature since <0.05 in many genetic studies.     

Shrinkage and Pretest Nonparametric Estimation of Regression Parameters from Censored Data with Multiple Observations at Each Level of Covariate

A simple linear regression model is considered where the independent variable assumes only a finite number of values and the response variable is randomly right censored. However, the censoring distribution may depend on the covariate values. A class of noniterative estimators for the slope parameter, namely, the noniterative unrestricted estimator, noniterative restricted estimator and noniterative improved pretest estimator are proposed. The asymptotic bias and mean squared errors of the proposed estimators are derived and compared. The relative dominance picture of the estimators is investigated. A simulation study is also performed to asses the properties of the various estimators for small samples.     

Estimating Ratio of Proportions Using Auxiliary Information

The problem of estimation of ratio of population proportions is considered and a difference-type estimator is proposed using auxiliary information. The bias and mean squared error of the proposed estimator is found and compared to the usual estimator and also to WYNN'S (1976) type estimator. An example is included for illustration.     

Estimation of the Density of G Given Observations of Y=G+E

We study the problem of estimating the density of a random variable G, given observations of a random variable Y = G + E. The random variable E is independent of G and its probability distribution function is considered as known. We build a family of estimators of the density of G using characteristic functions. We then derive a family of estimators of the density of Y based on the model for Y. The estimators are shown to be asymptotically unbiased and consistent. Simulations show that these estimators are better, as measured by integrated squared error, than the standard kernel estimators. Finally, we give an example of the use of this method for the detection of major genes in animal populations.     

A New Exponential Ratio-Type Estimator with Linear Combination of Two Auxiliary Variables

In sample surveys, it is usual to make use of auxiliary information to increase the precision of estimators. We propose a new exponential ratio-type estimator of a finite population mean using linear combination of two auxiliary variables and obtain mean square error (MSE) equation for proposed estimator. We find theoretical conditions that make proposed estimator more efficient than traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja and the estimator proposed by Abu-Dayeh et al. In addition, we support these theoretical results with the aid of two numerical examples.     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Penalized partial likelihood regression for right-censored data with bootstrap selection of the penalty parameter

The Cox proportional hazards model is often used for estimating the association between covariates and a potentially censored failure time, and the corresponding partial likelihood estimators are used for the estimation and prediction of relative risk of failure. However, partial likelihood estimators are unstable and have large variance when collinearity exists among the explanatory variables or when the number of failures is not much greater than the number of covariates of interest. A penalized (log) partial likelihood is proposed to give more accurate relative risk estimators. We show that asymptotically there always exists a penalty parameter for the penalized partial likelihood that reduces mean squared estimation error for log relative risk, and we propose a resampling method to choose the penalty parameter. Simulations and an example show that the bootstrap-selected penalized partial likelihood estimators can, in some instances, have smaller bias than the partial likelihood estimators and have smaller mean squared estimation and prediction errors of log relative risk. These methods are illustrated with a data set in multiple myeloma from the Eastern Cooperative Oncology Group.     

Network Structure and Biased Variance Estimation in Respondent Driven Sampling

This paper explores bias in the estimation of sampling variance in Respondent Driven Sampling (RDS). Prior methodological work on RDS has focused on its problematic assumptions and the biases and inefficiencies of its estimators of the population mean. Nonetheless, researchers have given only slight attention to the topic of estimating sampling variance in RDS, despite the importance of variance estimation for the construction of confidence intervals and hypothesis tests. In this paper, we show that the estimators of RDS sampling variance rely on a critical assumption that the network is First Order Markov (FOM) with respect to the dependent variable of interest. We demonstrate, through intuitive examples, mathematical generalizations, and computational experiments that current RDS variance estimators will always underestimate the population sampling variance of RDS in empirical networks that do not conform to the FOM assumption. Analysis of 215 observed university and school networks from Facebook and Add Health indicates that the FOM assumption is violated in every empirical network we analyze, and that these violations lead to substantially biased RDS estimators of sampling variance. We propose and test two alternative variance estimators that show some promise for reducing biases, but which also illustrate the limits of estimating sampling variance with only partial information on the underlying population social network.     

A restricted maximum likelihood estimator for truncated height samples

A restricted maximum likelihood (ML) estimator is presented and evaluated for use with truncated height samples. In the common situation of a small sample truncated at a point not far below the mean, the ordinary ML estimator suffers from high sampling variability. The restricted estimator imposes an a priori value on the standard deviation and freely estimates the mean, exploiting the known empirical stability of the former to obtain less variable estimates of the latter. Simulation results validate the conjecture that restricted ML behaves like restricted ordinary least squares (OLS), whose properties are well established on theoretical grounds. Both estimators display smaller sampling variability when constrained, whether the restrictions are correct or not. The bias induced by incorrect restrictions sets up a decision problem involving a bias-precision tradeoff, which can be evaluated using the mean squared error (MSE) criterion. Simulated MSEs suggest that restricted ML estimation offers important advantages when samples are small and truncation points are high, so long as the true standard deviation is within roughly 0.5 cm of the chosen value.     

Optimizing performance of nonparametric species richness estimators under constrained sampling

Understanding the functional relationship between the sample size and the performance of species richness estimators is necessary to optimize limited sampling resources against estimation error. Nonparametric estimators such as Chao and Jackknife demonstrate strong performances, but consensus is lacking as to which estimator performs better under constrained sampling. We explore a method to improve the estimators under such scenario. The method we propose involves randomly splitting species‐abundance data from a single sample into two equally sized samples, and using an appropriate incidence‐based estimator to estimate richness. To test this method, we assume a lognormal species‐abundance distribution (SAD) with varying coefficients of variation (CV), generate samples using MCMC simulations, and use the expected mean‐squared error as the performance criterion of the estimators. We test this method for Chao, Jackknife, ICE, and ACE estimators. Between abundance‐based estimators with the single sample, and incidence‐based estimators with the split‐in‐two samples, Chao2 performed the best when CV < 0.65, and incidence‐based Jackknife performed the best when CV > 0.65, given that the ratio of sample size to observed species richness is greater than a critical value given by a power function of CV with respect to abundance of the sampled population. The proposed method increases the performance of the estimators substantially and is more effective when more rare species are in an assemblage. We also show that the splitting method works qualitatively similarly well when the SADs are log series, geometric series, and negative binomial. We demonstrate an application of the proposed method by estimating richness of zooplankton communities in samples of ballast water. The proposed splitting method is an alternative to sampling a large number of individuals to increase the accuracy of richness estimations; therefore, it is appropriate for a wide range of resource‐limited sampling scenarios in ecology.     

Reliability of Confidence Interval Estimators under Various Nested Design Parental Sample Sizes

The coverage probabilities of several confidence limit estimators of genetic parameters, obtained from North Carolina I designs, were assessed by means of Monte Carlo simulations. The reliability of the estimators was compared under three different parental sample sizes. The coverage of confidence intervals set on the Normal distribution, and using standard errors either computed by the “delta” method or derived using an approximation for the variance of a variance component estimated by means of a linear combination of mean squares, was affected by the number of males and females included in the experiment. The “delta” method was found to provide reliable standard errors of the genetic parameters only when at least 48 males were each mated to six different females randomly selected from the reference population. Formulae are provided for obtaining “delta” method standard errors, and appropriate statistical software procedures are discussed. The error rates of confidence limits based on the Normal distribution and using standard errors obtained by an approximation for the variance of a variance component varied widely. The coverage of F-distribution confidence intervals for heritability estimates was not significantly affected by parental sample size and consistently provided a mean coverage near the stated coverage. For small parental sample sizes, confidence intervals for heritability estimates should be based on the F-distribution.     

Influence functions and robust Bayes and empirical Bayes small area estimation

We introduce new robust small area estimation procedures basedon area-level models. We first find influence functions correspondingto each individual area-level observation by measuring the divergencebetween the posterior density functions of regression coefficientswith and without that observation. Next, based on these influencefunctions, properly standardized, we propose some new robustBayes and empirical Bayes small area estimators. The mean squarederrors and estimated mean squared errors of these estimatorsare also found. A small simulation study compares the performanceof the robust and the regular empirical Bayes estimators. Whenthe model variance is larger than the sample variance, the proposedrobust empirical Bayes estimators are superior.     

Inverse Adaptive Cluster Sampling

Consider a population in which the variable of interest tends to be at or near zero for many of the population units but a subgroup exhibits values distinctly different from zero. Such a population can be described as rare in the sense that the proportion of elements having nonzero values is very small. Obtaining an estimate of a population parameter such as the mean or total that is nonzero is difficult under classical fixed sample-size designs since there is a reasonable probability that a fixed sample size will yield all zeroes. We consider inverse sampling designs that use stopping rules based on the number of rare units observed in the sample. We look at two stopping rules in detail and derive unbiased estimators of the population total. The estimators do not rely on knowing what proportion of the population exhibit the rare trait but instead use an estimated value. Hence, the estimators are similar to those developed for poststratification sampling designs. We also incorporate adaptive cluster sampling into the sampling design to allow for the case where the rare elements tend to cluster within the population in some manner. The formulas for the variances of the estimators do not allow direct analytic comparison of the efficiency of the various designs and stopping rules, so we provide the results of a small simulation study to obtain some insight into the differences among the stopping rules and sampling approaches. The results indicate that a modified stopping rule that incorporates an adaptive sampling component and utilizes an initial random sample of fixed size is the best in the sense of having the smallest variance.