Confidence intervals for quantile estimation using Jackknife techniques

We consider the inference on quantiles, Q _y (β), with jackknife techniques, in finite populations of a variable, Y, using the quantile information on an auxiliary variable, X. Jackknife techniques are applied to estimate quantiles and the behaviour of these estimators is analyzed. Their properties are studied for simple random sampling. We also examine the confidence intervals obtained with jackknife variances.     

On the Bias of the Generalized Regression Estimator in Survey Sampling

It is well known that the generalized regression (GREG) estimator of the finite population total is asymptotically unbiased. Consequently, bias is negligible when the sample size is large. But the magnitude of the bias is not known, if we are estimating small areas or operating with small samples. Furthermore, beside the sample size, the bias depends on the auxiliary variables, on their relation to the study variable and on the sampling design. In small samples it is important to know sources of the bias and in some cases to use a bias-corrected regression estimator. The aim of the present paper is to derive approximate bias expressions of the GREG estimator under different population models and different sampling designs to study the magnitude of the bias.      

Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we consider ranked set sampling, in which ranking of units are done based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. We then estimate the mean of the study variate Y by the BLUE based on the measurements made on the units of the ranked set sampling regarding the study variable Y, when (X ,Y) follows a Morgenstern type bivariate exponential distribution. We then consider unbalanced multistage ranked set sampling and estimate the mean of the study variate Y by the BLUE based on the observations made on the units of multistage ranked set sample regarding the study variable Y. Efficiency comparison is also made on all estimators considered in this work.     

Ratio methods to the mean estimation with known quantiles

We address the problem of estimating the finite population mean in survey sampling, by exploiting any available auxiliary information in order to increase the precision of classical estimators. The idea is to use any population quantiles of the available auxiliary variables which are known in many real situation from census, administrative files, etc. This is achieved using these known quantities in the construction of the estimators, by modifying the usual ratio estimation methods and afterwards defining a general class of exponentiation ratio estimators. The advantages of the proposed estimators are demonstrated using theoretical asymptotic tools and through a simulation study.     

A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finite-population distribution function using two-phase sampling is given, and its properties are investigated.      

On the Edgeworth expansion and the <Emphasis Type="Italic">M</Emphasis> out of <Emphasis Type="Italic">N</Emphasis> bootstrap accuracy for a Studentized trimmed mean

We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].      

An exact bootstrap for variance of finite-population <Emphasis Type="BoldItalic">L</Emphasis>-statistic

We give an exact formula of a finite-population bootstrap variance estimator for a general class of L-statistic. It is aimed to reduce the computational burden and to eliminate the approximation error, typically present in resampling approximations based on simulation. In the case of the classical nonparametric Efron bootstrap, a similar formula was shown by Hutson and Ernst [A.D. Hutson and M.D. Ernst, The exact bootstrap mean and variance of an L-estimator, J. R. Stat. Soc., Ser. B, 62:89–94, 2000].     

Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

Asymptotic properties of sample quantiles from a finite population

In this paper we consider the problem of estimating quantiles of a finite population of size N on the basis of a finite sample of size n selected without replacement. We prove the asymptotic normality of the sample quantile and show that the scaled variance of the sample quantile converges to the asymptotic variance under a slight moment condition. We also consider the performance of the bootstrap in this case, and find that the usual (Efron’s) bootstrap method fails to be consistent, but a suitably modified version of the bootstrapped quantile converges to the same asymptotic distribution as the sample quantile. Consistency of the modified bootstrap variance estimate is also proved under the same moment conditions.     

Bootstrap method and empirical process

In this paper we consider the sampling properties of the bootstrap process, that is, the empirical process obtained from a random sample of size n (with replacement) of a fixed sample of size n of a continuous distribution. The cumulants of the bootstrap process are given up to the order n ^–1 and their unbiased estimation is discussed. Furthermore, it is shown that the bootstrap process has an asymptotic minimax property for some class of distributions up to the order n ^–1/2.     

Optimum stratification with ratio and regression methods of estimation

Summary The paper considers the problem of optimum stratification on an auxiliary variablex when the information on the auxiliary variablex is also used to estimate the population mean using ratio or regression methods of estimation. Assuming the form of the regression of the estimation variabley on the auxiliary variablex as also the form of the conditional variance function V(y/x), the problem of determining optimum strata boundaries (OSB) is shown to be a particular case of optimum stratification on the auxiliary variable for stratified simple random sampling estimate. A numerical investigation has also been made to study the amount of gain in efficiency that can be brough about by stratifying the population.     

Bootstrap for the conditional distribution function with truncated and censored data

We propose a resampling method for left truncated and right censored data with covariables to obtain a bootstrap version of the conditional distribution function estimator. We derive an almost sure representation for this bootstrapped estimator and, as a consequence, the consistency of the bootstrap is obtained. This bootstrap approximation represents an alternative to the normal asymptotic distribution and avoids the estimation of the complicated mean and variance parameters of the latter.     

Estimation of parameters in a two-parameter exponential distribution using ranked set sample

In situations where the experimental or sampling units in a study can be easily ranked than quantified, McIntyre (1952,Aust. J. Agric. Res.,3, 385–390) proposed that the mean ofn units based on aranked set sample (RSS) be used to estimate the population mean, and observed that it provides an unbiased estimator with a smaller variance compared to a simple random sample (SRS) of the same sizen. McIntyre's concept ofRSS is essentially nonparametric in nature in that the underlying population distribution is assumed to be completely unknown. In this paper we further explore the concept ofRSS when the population is partially known and the parameter of interest is not necessarily the mean. To be specific, we address the problem of estimation of the parameters of a two-parameter exponential distribution. It turns out that the use ofRSS and its suitable modifications results in much improved estimators compared to the use of aSRS.     

随机删失下概率密度核估计的光滑Bootstrap逼近

本文研究随机删失概率密度估计的光ｂｏｏｔｓｔｒａｐ逼近。给出了光滑ｂｏｏｔｓｔｒａｐ逼近成立的充分条件，并证明了概率密度的光滑ｂｏｏｔｓｔｒａｐ估计方差几乎处处收敛到概率密度核估计的渐近方差。     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

A new resampling method for sampling designs without replacement: the doubled half bootstrap

A new and very fast method of bootstrap for sampling without replacement from a finite population is proposed. This method can be used to estimate the variance in sampling with unequal inclusion probabilities and does not require artificial populations or utilization of bootstrap weights. The bootstrap samples are directly selected from the original sample. The bootstrap procedure contains two steps: in the first step, units are selected once with Poisson sampling using the same inclusion probabilities as the original design. In the second step, amongst the non-selected units, half of the units are randomly selected twice. This procedure enables us to efficiently estimate the variance. A set of simulations show the advantages of this new resampling method.     

Exact convergence rate of bootstrap quantile variance estimator

Summary It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n ^-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n ^-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n ^-2/5.     

基于轮换样本连续性调查的校准组合估计方法研究

校准估计是抽样调查中比较常用的一种利用辅助信息提高估计量精度的方法。回归组合估计量作为轮换样本连续性调查中使用的一种有效的估计量,是可以通过校准程序得到的。基于回归组合估计量和校准程序之间的关系,本文提出了轮换样本连续性抽样调查条件下的不同校准组合估计量及其方差估计。校准组合估计量的主要思想是在校准估计程序中将拼配样本和非拼配样本的辅助信息进行不同的组合利用。本文利用美国现时人口调查的微观数据进行数值模拟,来比较不同校准组合估计量的估计效率,模拟结果表明两步校准组合估计量和两步校准双组合估计量的表现相似,且估计精度都高于H-T估计量及回归组合估计量;而两步校准组合估计量由于其简便性更适合应用于实践中。最后以我国农村住户连续性抽样调查为例,设计一套符合我国实际的轮换样本连续性调查方案,并将提出的校准组合估计量运用于估计阶段,为中国政府统计调查提供一定的借鉴和参考.     

Risk Estimation for Classification Trees

This article is a study of techniques for bias reduction of estimates of risk both globally and within terminal nodes of CART^R classification trees. In Section 5.4 of Classification and Regression Trees, Leo Breiman presented an estimator that has two free parameters. An empirical Bayes method was put forth for estimating them. Here we explain why the estimator should be successful in the many examples for which it is. We give numerical evidence from simulations in the two-class case with attention to ordinary resubstitution and seven other methods of estimation. There are 14 sampling distributions, all but one simulated and the remaining concerning E. coli promoter regions. We report on varying minimum node sizes of the trees; prior probabilities and misclassification costs; and, when relevant, the numbers of bootstraps or cross-validations. A variation of Breiman's method in which repeated cross-validation is employed to estimate global rates of misclassification was the most accurate from among the eight methods. Exceptions are cases for which the Bayes risk of the Bayes rule is small. For them, either a local bootstrap .632 estimate or Breiman's method modified to use a bootstrap estimate of the global misclassification rate is most accurate.     

Bayesian variable sampling plan for the weibull distribution with type I censoring

In this article, we study a model of a single variable sampling plan with Type I censoring. Assume that the quality of an item in a batch is measured by a random variable which follows a Weibull distributionW (λ,m), with scale parameter λ and shape parameterm having a gamma-discrete prior distribution or σ=1/λ andm having an inverse gamma-uniform prior distribution. The decision function is based on the Kaplan-Meier estimator. Then, the explicit expressions of the Bayes risk are derived. In addition, an algorithm is suggested so that an optimal sampling plan can be determined approximately after a finite number of searching steps.