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.     

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.     

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.     

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.     

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.     

On Combination of Ratio and PPS Estimators

AGARWAL and KUMAR (1980) proposed an estimator, combining ratio and pps estimators of population mean and proved that the proposed estimator would always be better (in minimum mean square error sense) than the pps estimator or the ratio estimator under pps sampling scheme for optimum value of constant k (parameter). The optimum value of k is rarely known in practice, hence the alternative is to replace k from the sample-values. In this paper, an estimator depending on estimated optimum value of k based on sample-values, under pps sampling scheme is proposed and studied.     

Efficiency of the Sample Variance for the Lognormal Distribution

Two unbiased estimators T and ?? of the variance ? = var (Y) of a lognormal distribution are considered. Here T is the sample variance and ?? is the minimum variance unbiased estimator of ?. The values of the ratio E = 100 var (??)/var (T) are tabulated for some values of the sample size n and of the coefficient of variation δ.     

On an Unbiased Variance Estimator for the WILCOXON-MANN-WHITNEY-Statistic Based on Ranks

For some applications of the WILCOXON-MANN-WHITNEY-statistic its variance has to be estimated. So e.g. for the test of POTTHOFF (1963) to detect differences in medians of two symmetric distributions as well as for the computation of approximate, confidence bounds for the probability P(X₁ ≥ X₂), cf. GOVINDARAJULU (1968). In the present paper an easy to compute variance estimator is proposed which as only information uses the ranks of the data with the additional property that it is unbiased for the finite variance. Because of its invariance under any monotone transformation of the data its applicability is not confined to quantitative data. The estimator may be applied to ordinal data just as well. Some properties are discussed and a numerical example is given.     

Estimators Asymptotically Minimax in Wide Sense

Estimators of location are considered. Huber (1964) introduced estimators asymptotically minimax on the set ?? of all regular M-estimators, for a given contamination ε and for the set Q of all regular symmetric alternative data sources. We extend his concept by admitting arbitrary sets ?? of regular M-estimators and arbitrary sets Q or regular symmetric alternative sources, and also by replacing the singletons [ε] ? (0, 1) by arbitrary subsets ?? ? (0, 1). The resulting estimator cannot in general be evaluated explicitly. But for finite T it exists and, if ?? and Q are finite too, it may be chosen by a computer. This extra burden is justified in some cases since more than 100% relative efficiency gain against all Huber's Hk is achievable in this manner. Such gains are achieved for a nontrivial family Q by the estimator proposed in Vajda (1984), with redescending influence curve, which is shown to be asymptotically minimax in wide sense.     

Sufficiency of the number of segregating sites in the limit under finite-sites mutation

We show that the number of segregating sites is a sufficient statistic for the scaled mutation parameter (θ) in the limit as the number of sites tends to infinity and there is free recombination between sites. We assume that the mutation parameter at each site tends to zero such than the total mutation parameter (θ) is constant in the limit. Our results show that Watterson’s estimator is the maximum likelihood estimator in this case, but that it estimates a composite parameter which is different for different mutation models. Some of our results hold when recombination is limited, because Watterson’s estimator is an unbiased, method-of-moments estimator regardless of the recombination rate. The quantity it estimates depends on the details of how mutations occur at each site.     

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.     

MINQUE and ANOVA Estimator for One-way Classification - a Risk Comparison

For the one-way classification in unbalanced case MINQUEstimator for components of variance are given in a more explicit form than it is done in the paper from C. R. RAO (1971). By means of the risk functions we compare MINQUE and ANOVA estimator. For given n_j-patterns angular ranges in the positive quadrant are given where MINQUE is better than ANOVA estimator. A special n_j-pattern and one parameter δ₀ is found for which MINQUE is uniformly better than ANOVA. Limit values are given for MINQUE for δ₀ = ∞ and δ₀ = 0 and their relations to the ANOVA estimator are considered. The coincidence between MINQUE and ANOVA for balanced case is verified. Extensive numerical studies for real data are carried out which stimulated the search for a fixpoint δ as a point for which the distance to the initial parameter δ₀ is as small as possible.     

A Class of Ratio Cum Product-Type Estimator in Double Sampling

A class of ratio cum product-type estimator is proposed in case of double sampling in the present paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and a good range of α-values, it is found that the proposed estimator is more efficient than the set of estimator viz., simple mean estimator, usual ratio and product estimators, SRIVASTAVA 's estimator (1967), CHAKARBARTY 's estimator and product-type estimator, which are in fact the particular cases of it. The proposed estimator is as efficient as linear regression estimator in double sampling at optimum value of α.     

Estimating effective population size from linkage disequilibrium: severe bias in small samples

Effective population size (N _e) is a central concept in evolutionary biology and conservation genetics. It predicts rates of loss of neutral genetic variation, fixation of deleterious and favourable alleles, and the increase of inbreeding experienced by a population. A method exists for the estimation of N _e from the observed linkage disequilibrium between unlinked loci in a population sample. While an increasing number of studies have applied this method in natural and managed populations, its reliability has not yet been evaluated. We developed a computer program to calculate this estimator of N _e using the most widely used linkage disequilibrium algorithm and used simulations to show that this estimator is strongly biased when the sample size is small (<‰100) and below the true N _e. This is probably due to the linkage disequilibrium generated by the sampling process itself and the inadequate correction for this phenomenon in the method. Results suggest that N _e estimates derived using this method should be regarded with caution in many cases. To improve the method’s reliability and usefulness we propose a way to determine whether a given sample size exceeds the population N _e and can therefore be used for the computation of an unbiased estimate.     

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．     

Minimum Variance Unbiased Estimation in a Bivariate Modified Power Series Distribution

The functional form of the class of Bivariate Modified Power Series Distributions is considered. The probabilities are functions of two unknown parameters ?₁ and ?_2. The necessary and sufficient conditions for the existence of a minimum variance unbiased estimator for a parametric function of ?₁ and ?₂ are given.     

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.     

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.     

A Comparison of Nonparametric Error Rate Estimation Methods in Classification Problems

Assessment of the misclassification error rate is of high practical relevance in many biomedical applications. As it is a complex problem, theoretical results on estimator performance are few. The origin of most findings are Monte Carlo simulations, which take place in the “normal setting”: The covariables of two groups have a multivariate normal distribution; The groups differ in location, but have the same covariance matrix and the linear discriminant function LDF is used for prediction. We perform a new simulation to compare existing nonparametric estimators in a more complex situation. The underlying distribution is based on a logistic model with six binary as well as continuous covariables. To study estimator performance for varying true error rates, three prediction rules including nonparametric classification trees and parametric logistic regression and sample sizes ranging from 100‐1,000 are considered. In contrast to most published papers we turn our attention to estimator performance based on simple, even inappropriate prediction rules and relatively large training sets. For the major part, results are in agreement with usual findings. The most strikingly behavior was seen in applying (simple) classification trees for prediction: Since the apparent error rate Êrr.app is biased, linear combinations incorporating Êrr.app underestimate the true error rate even for large sample sizes. The .632+ estimator, which was designed to correct for the overoptimism of Efron's .632 estimator for nonparametric prediction rules, performs best of all such linear combinations. The bootstrap estimator Êrr.B0 and the crossvalidation estimator Êrr.cv, which do not depend on Êrr.app, seem to track the true error rate. Although the disadvantages of both estimators – pessimism of Êrr.B0 and high variability of Êrr.cv – shrink with increased sample sizes, they are still visible. We conclude that for the choice of a particular estimator the asymptotic behavior of the apparent error rate is important. For the assessment of estimator performance the variance of the true error rate is crucial, where in general the stability of prediction procedures is essential for the application of estimators based on resampling methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Use of Transformed Auxiliary Variable in Estimating the Finite Population Mean

For estimating the finite population mean Y- of the study character y, an estimator using a transformed auxiliary variable has been defined. The bias and mean-squared error (MSE) of the proposed estimator have been obtained. The regions of preference have been obtained under which it is better than usual unbiased estimator y-, the ratio estimator y-_R = y-X-/x-, Sisodia and Dwivedi (1981) estimator y-_s = y-(X- + C_x)/(x- + C_x) and Singh and Kakran (1993) estimator y-_k = y[X- + β₂(x)]/[x- + β₂(x)]. An empirical study has been carried out to demonstrate the superiority of the suggested estimator over the others.