A Class of Almost Unbiased Estimators for Finite Populations Mean Using Two Auxiliary Variables

For estimating finite population mean -Y₀ of study character y₀, a class of almost unbiased estimators applying jackknife technique envisaged by Quenouille (1956) is derived. Optimum unbiased estimator (OUE) is also investigated with its variance formula. An empirical study is carried out to demonstrate the performance of the constructed estimator over the usual unbiased estimator, Srivastava (1965), Singh (1967), Singh and Biradar (1992), Tracy , Singh , and Singh (1996) and other almost unbiased estimators.     

A note on robust variance estimation for cluster-correlated data

There is a simple robust variance estimator for cluster-correlated data. While this estimator is well known, it is poorly documented, and its wide range of applicability is often not understood. The estimator is widely used in sample survey research, but the results in the sample survey literature are not easily applied because of complications due to unequal probability sampling. This brief note presents a general proof that the estimator is unbiased for cluster-correlated data regardless of the setting. The result is not new, but a simple and general reference is not readily available. The use of the method will benefit from a general explanation of its wide applicability.     

On Use of Distinct Respondents in Randomized Response Surveys

It is well known for direct response surveys (DR), where the responses are obtained from the respondents directly, that the sample mean, based on distinct units of a simple random sample selected with replacement (SRSWR) method, is more efficient than the sample mean based on all the units including repetition. In this paper, it is shown that a linear unbiased estimator based on distinct units is inadmissible for estimating a finite population mean when the sample is selected by an arbitrary with replacement (WR) sampling scheme and the responses are obtained independently by some RR technique. Efficiencies for a few linear unbiased estimators are compared under SRSWR sampling.     

Some Early Developments in Ratio Estimation

John Graunt (1662) was the first to estimate the ratio y/x where y represents the total population and x the known total number of registered births in the same areas during the preceding year. About 1765 Messance (Stephan, 1948) and Moheau (1778) published very carefully prepared estimates for France based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample. Laplace (1786) prepared similar estimates in 1802 based on a two-stage sampling plan. Recently Hansen and Hurwitz (1943) showed that the ratio estimate (y_i/n_i)X of Y is unbiased where all x_i's are known and the n^th cluster is selected with p.p.s. More recently Hájek (1949), Lahiri (1951), Midzuno (1952) and Sen (1952) developed independently the sampling of n clusters with p.p.s to the totals of the sizes of the sample clusters S(x_i). Des Raj (1954) and Sen (1952, 1953) gave unbiased estimate of the variance of the estimator which was generally non-negative for samples with smaller probabilities. Rao and Vijayan (1977) gave an unbiased estimator which is non-negative for samples with larger probabilities. Hájek (1949) provided an almost unbiased estimator of the variance of the estimator. The paper discusses situations where Hájek's estimator of variance should be preferred to the Rao-Vijayan estimator and vice versa.     

Simultaneously model-unbiased, design-unbiased estimation

This paper proposes a class of inferential procedures (incorporating both design and estimation elements) that yield estimates of means that are simultaneously model unbiased and design unbiased. Classical regression procedures yield conditionally unbiased estimators for the mean (conditioning on the model and choice of observation points). In contrast, design-based methods yield estimators that are unconditionally unbiased on matter what the form of the underlying model. Variance properties of the proposed class are examined, and applications to bioavailability, water quality from mine run-off, and finite population regression estimation are considered. The proposed procedures perform well, especially in the typical case where a model is only approximately correct.     

Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

A biased but simple and consistent estimator of the parameter ? has been obtained for the normal distribution N(?, a?₂), ?>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).     

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.     

Logistic regression when covariates are random effects from a non‐linear mixed model

In many studies, the association of longitudinal measurements of a continuous response and a binary outcome are often of interest. A convenient framework for this type of problems is the joint model, which is formulated to investigate the association between a binary outcome and features of longitudinal measurements through a common set of latent random effects. The joint model, which is the focus of this article, is a logistic regression model with covariates defined as the individual‐specific random effects in a non‐linear mixed‐effects model (NLMEM) for the longitudinal measurements. We discuss different estimation procedures, which include two‐stage, best linear unbiased predictors, and various numerical integration techniques. The proposed methods are illustrated using a real data set where the objective is to study the association between longitudinal hormone levels and the pregnancy outcome in a group of young women. The numerical performance of the estimating methods is also evaluated by means of simulation.     

Comparing Stochastically Restricted Estimators in a Linear Regression Model

The minimum dispersion linear unbiased estimators of the vector of parameters in a linear regression model are compared when the parameters of the model are subject to stochastic linear restrictions with different dispersion matrices of the disturbances involved in them.     

Estimating risk difference in multicenter studies under baseline-risk heterogeneity

In this paper, we consider the case of efficient estimation of the risk difference in a multicenter study allowing for baseline heterogeneity. We consider the optimally weighted estimator for the common risk difference and show that this estimator has considerable bias when the true weights (which are inversely proportional to the variances of the center-specific risk difference estimates) are replaced by their sample estimates. In addition, we propose a new estimator for this situation of the Mantel-Haenszel type that is unbiased and, in addition, has a smaller variance for small sample sizes within the study centers. Simulations illustrate these findings.     

On the equivalence of meta-analysis using literature and using individual patient data

When data come from several independent studies for the purpose of estimating treatment control differences, meta-analysis can be carried out either on the best linear unbiased estimators computed from each study or on the pooled individual patient data modelled as a two-way model without interaction, where the two factors represent the different studies and the different treatments. Assuming that observations within and between studies are independent having a common variance, Olkin and Sampson (1998) have obtained the surprising result that the two meta-analytic procedures are equivalent, i.e., they both produce the same estimator. In this article, the same equivalence is established for the two-way fixed-effects model without interaction with the only assumption that the observations across studies be independent. A consequence of the equivalence result is that, regardless of the covariance structure, it is possible to get an explicit representation for the best linear unbiased estimator of any vector of treatment contrasts in a two-way fixed-effects model without interaction as long as the studies are independent. Another interesting consequence is that, for the purpose of best linear unbiased estimation, an unbalanced two-way fixed-effects model without interaction can be treated as several independent unbalanced one-way models, regardless of the covariance structure, when the studies are independent.     

两阶抽样中比率型估计量的平均精度

本文给出了两阶抽样中总体均值的比率型估计量的平均精度,它当样本容量充分大时主项不劣于无偏估计量的平均精度．     

Estimation of Finite Population Variance Using Ratio and Product Methods of Estimation

For estimating finite population variance σ_y² of a character y under our study, estimators using auxiliary information on a character x in the form of ratio, product, ratio-type or product-type estimators have been suggested, and their comparative study with the conventional unbiased estimator s_y² of σ_y² has been made in simple random sampling with replacement. A generalized estimator representing a class of estimators for the finite populations variance, has also been studied.     

Unbiasedness of Iterated Aitken Estimators

The iterated Aitken estimator of the parameter vector of the general linear model is shown to be unbiased on the assumption that the errors underlying the model follow a symmetric probability law with mean zero.     

Unbiasedness of the Estimator of the Function of Expected Value in the Mixed Linear Model

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two-stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method. In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation-invariant and are even functions of the observation vector.     

The Utilization of Kurtosis in the Estimation of the Parameters of the One‐Way Random Effects Model

This paper deals with the problem of estimating the components of the variance of one‐way random effects model under non‐normality situation using a prior knowledge of coefficient of kurtosis. We have suggested two classes of estimators and for the within and between variances respectively. Optimum estimators in the classes of and are identified with their mean squared errors formulae and compared with that of usual ANOVA unbiased and Shoukri , Tracy and Mian 's (1990) estimators. It is found that the proposed estimators are more efficient than the ANOVA unbiased estimators and Shoukri , Tracy and Mian (1990) estimators.     

New Ranked Set Sampling for One‐Parameter Family of Distributions

Bhoj (1997c) proposed a new ranked set sampling (NRSS) procedure for a specific two‐parameter family of distributions when the sample size is even. This NRSS procedure can be applied to one‐parameter family of distributions when the sample size is even. However, this procedure cannot be used if the sample size is odd. Therefore, in this paper, we propose a modified version of the NRSS procedure which can be used for one‐parameter distributions when the sample size is odd. Simple estimator for the parameter based on proposed NRSS is derived. The relative precisions of this estimator are higher than those of other estimators which are based on other ranked set sampling procedures and the best linear unbiased estimator using all order statistics.     

Stabilization Coefficient of Random Variable

Coefficient of variation, standard deviation divided by mean, has some essential defects. Its density, expectation and variance are too complex to make the statistical inference for such a coefficient. The definition of stabilization coefficient is just the reciprocal of variation coefficient, mean divided by standard deviation. Such a coefficient has a simple expectation and a simple variance, and is an asymptotically unbiased estimator and a consistent estimator of its true value. Furthermore, coefficient of stabilization has an asymptotic normality. Due to its statistical advantages, coefficient of stabilization is easy to be tested statistically. In some applied fields, usually, there is an increasing standard deviation accompanying an increasing mean. Coefficient of stabilization can be practically used for some comparison studies in such fields. Illustrations about comparing microorganism strains are given in this paper. The robustness of stabilization coefficient is satisfactory.     

A Note on Balanced Sampling and Robust Genetic Predictors

The concept of balanced sampling is applied to prediction in finite samples using model based inference procedures. Necessary and sufficient conditions are derived for a general linear model with arbitrary covariance structure to yield the expansion estimator as the best linear unbiased predictor for the mean. The analysis is extended to produce a robust estimator for the mean squared error under balanced sampling and the results are discussed in the context of statistical genetics where appropriate sampling produces simple efficient and robust genetic predictors free from unnecessary genetic assumptions.     

Unbiased relatedness estimation in structured populations

Knowledge of the genetic relatedness between individuals is important in many research areas in quantitative genetics, conservation genetics, forensics, evolution, and ecology. In the absence of pedigree records, relatedness can be estimated from genetic marker data using a number of estimators. These estimators, however, make the critical assumption of a large random mating population without genetic structures. The assumption is frequently violated in the real world where geographic/social structures or nonrandom mating usually lead to genetic structures. In this study, I investigated two approaches to the estimation of relatedness between a pair of individuals from a subpopulation due to recent common ancestors (i.e., relatedness is defined and measured with the current focal subpopulation as reference). The indirect approach uses the allele frequencies of the entire population with and without accounting for the population structure, and the direct approach uses the allele frequencies of the current focal subpopulation. I found by simulations that currently widely applied relatedness estimators are upwardly biased under the indirect approach, but can be modified to become unbiased and more accurate by using Wright's F(st) to account for population structures. However, the modified unbiased estimators under the indirect approach are clearly inferior to the unmodified original estimators under the direct approach, even when small samples are used in estimating both allele frequencies and relatedness.