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.     

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.     

基于样本均值的最优无重叠k-序对排序集抽样

本文提出了无重叠κ-序对排序集抽样方法,即在每个排序集中对κ-序对个体进行观测,并且不同的排序集的κ-序对之间没有任何重复.我们首先探究了此抽样方法得到的样本均值的有效性随每个排序集中κ-序对个体间的相关性变化的趋势.κ-序对个体间的相关性越强,样本均值的有效性损失越大.本文的目的是找到无重叠κ-序对排序集抽样方法中κ-序对分配的最优方案从而使样本均值的有效性损失最小,并证明了最优的无重叠κ-序对排序集抽样比广义排序集抽样以及简单随机抽样更有效.尽管无重叠κ-序对排序集抽样方法的统计效率低于经典的排序集抽样,但是在成本模型下,最优的无重叠κ-序对排序集抽样方法可以比经典的排序集抽样更有效.     

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.      

Estimation of the Common Mean of a Bivariate Normal Population

In this paper we discuss the problem of estimating the common mean of a bivariate normal population based on paired data as well as data on one of the marginals. Two double sampling schemes with the second stage sampling being either a simple random sampling (SRS) or a ranked set sampling (RSS) are considered. Two common mean estimators are proposed. It is found that under normality, the proposed RSS common mean estimator is always superior to the proposed SRS common mean estimator and other existing estimators such as the RSS regression estimator proposed by Yu and Lam (1997, Biometrics, 53, 1070–1080). The problem of estimating the mean Reid Vapor Pressure (RVP) of regular gasoline based on field and laboratory data is considered.     

On the Ranked-Set Sampling M-Estimates for Symmetric Location Families

The ranked-set sampling (RSS) is applicable in practical problems where the variable of interest for an observed item is costly or time-consuming but the ranking of a set of items according to the variable can be easily done without actual measurement. In this article, the M-estimates of location parameters using RSS data are studied. We deal mainly with symmetric location families. The asymptotic properties of M-estimates based on ranked-set samples are established. The properties of unbalanced ranked-set sample M-estimates are employed to develop the methodology for determining optimal ranked-set sampling schemes. The asymptotic relative efficiencies of ranked-set sample M-estimates are investigated. Some simulation studies are reported.     

Proportion estimation in ranked set sampling in the presence of tie information

Ranked set sampling (RSS) is a statistical technique that uses auxiliary ranking information of unmeasured sample units in an attempt to select a more representative sample that provides better estimation of population parameters than simple random sampling. However, the use of RSS can be hampered by the fact that a complete ranking of units in each set must be specified when implementing RSS. Recently, to allow ties declared as needed, Frey (Environ Ecol Stat 19(3):309–326, 2012) proposed a modification of RSS, which is to simply break ties at random so that a standard ranked set sample is obtained, and meanwhile record the tie structure for use in estimation. Under this RSS variation, several mean estimators were developed and their performance was compared via simulation, with focus on continuous outcome variables. We extend the work of Frey (2012) to binary outcomes and investigate three nonparametric and three likelihood-based proportion estimators (with/without utilizing tie information), among which four are directly extended from existing estimators and the other two are novel. Under different tie-generating mechanisms, we compare the performance of these estimators and draw conclusions based on both simulation and a data example about breast cancer prevalence. Suggestions are made about the choice of the proportion estimator in general.     

Order restricted randomized designs for control versus treatment comparison

We develop a method of randomizing units to treatments that relies on subjective judgement or on possible coarse modeling to produce restrictions on the randomization. The procedure thus fits within the general framework of ranked set sampling. However, instead of selecting a single unit from each set for full measurement, all units within a set are used. The units within a set are assigned to different treatments. Such an assignment translates the positive dependence among units within a set into a reduction in variation of contrasting features of the treatments. A test for treatment versus control comparison, with controlled familywise error rate, is developed along with the associated confidence intervals. The new procedure is shown to be superior to corresponding procedures based on completely randomized or ranked set sample designs. The superiority appears both in asymptotic relative efficiency and in power for finite sample sizes. Importantly, this test does not rely on perfect rankings; rather, the information in the data on the quality of rankings is exploited to maintain the level of the test when rankings are imperfect. The asymptotic relative efficiency of the test is not affected by estimation of the quality of rankings, and the finite sample performance is only mildly affected.     

基于有序抽样样本的参数的极大似然估计的性质

有序抽样是一种新的抽样方法 ,与简单随机抽样方法相比它具有很多很好的性质 .本文讨论了在有序抽样样本下的参数的极大似然估计的性质 .     

Rounded data analysis based on multi-layer ranked set sampling

Observations of sampling are often subject to rounding, but are modeled as though they were unrounded. This paper examines the impact of rounding errors on parameter estimation with multi-layer ranked set sampling. It shows that the rounding errors seriously distort the behavior of covariance matrix estimate, and lead to inconsistent estimation. Taking this into account, we present a new approach to implement the estimation for this model, and further establish the strong consistency and asymptotic normality of the proposed estimators. Simulation experiments show that our estimates based on rounded multi-layer ranked set sampling are always more efficient than those based on rounded simple random sampling.     

Exact two-sample nonparametric test for quantile difference between two populations based on ranked set samples

Creation of a ranked set sample, by its nature, involves judgment ranking error within set units. This ranking error usually distorts statistical inference of the population characteristics. Tests may have inflated sizes, confidence intervals may have incorrect coverage probabilities, and the estimators may become biased. In this paper, we develop an exact two-sample nonparametric test for quantile shift between two populations based on ranked set samples. This test is based on two independent exact confidence intervals for the quantile of interest corresponding to the two populations and rejects the null hypothesis of equal quantiles if these intervals are disjoint. It is shown that a pair of 83 and 93% confidence intervals provide a 5 and 1% test for the equality of quantiles. The proposed test is calibrated for the effect of judgment ranking error so that the test has the correct size even under a wide range of judgment ranking errors. A small scale simulation study suggests that the test performs quite well for cycle sizes as small as 2.     

基于中位数排序集抽样的非参数估计

本文提出中位数排序集抽样下总体中位数的非参数估计,证明了这种估计具有强相合性和渐近正态性,并系统验证了新估计量的功效一致优于排序集抽样下和简单随机抽样下总体中位数的估计量。最后,我们对针叶树的一组真实数据进行了实际应用。     

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.     

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.     

优良抽样设计下总体均值的比率估计

当研究目标的实际测量具有不可修复的破坏性或耗资巨大时,有效的抽样设计将是一项重要的研究课题.在统计推断方面,排序集抽样被视为一种更为有效的收集数据的方式.极值排序集抽样(ERSS)是一种改进的排序集抽样.文章在ERSS下研究了总体均值的比率估计.以正态分布为例,比较了简单随机抽样和ERSS下比率估计的相对效率.数值结果表明ERSS下的比率估计优于简单随机抽样下的比率估计.     

序集抽样中$M$估计分布的随机加权逼近

序集抽样是一种适用于准确测量花费太高而排序费用可以忽略不记时的一种抽样方法.讨论了序集抽样下的对于一般分布族M估计的相合性和渐近正态性并且通过随机加权的方法来估计M估计的分布.     

Bootstrap Algorithms for Risk Models with Auxiliary Variable and Complex Samples

Resampling methods are often invoked in risk modelling when the stability of estimators of model parameters has to be assessed. The accuracy of variance estimates is crucial since the operational risk management affects strategies, decisions and policies. However, auxiliary variables and the complexity of the sampling design are seldom taken into proper account in variance estimation. In this paper bootstrap algorithms for finite population sampling are proposed in presence of an auxiliary variable and of complex samples. Results from a simulation study exploring the empirical performance of some bootstrap algorithms are presented.      

Continuous sampling plans for Markov-dependent production processes under limited inspection capacity

A continuous sampling plan is a set of rules that provide a given Average Outgoing Quality (AOQ), ideally with the minimum of effort (as measured by the Average Fraction Inspected, or AFI). Most such plans are based on the assumption that the quality (either defective or not) of successive production units is uncorrelated. In this paper, we explore the impact of correlation in the production process on the design of a sampling plan when it is not possible to inspect long runs of production unit-by-unit. We shall generalize Dodge's continuous sampling plan on two counts, replacing Level 1 100% inspection by 100f_o% inspection, and considering the production process to be Markov dependent instead of consisting of independent Bernoulli trials. We derive formulae for the AOQ and AFI, and consider how best to choose the sampling plan parameters in the presence of nonzero correlation.     

产品可靠寿命的最优排序集挑选设计

针对产品可靠寿命的估计问题,构造了基于排序集挑选样本的非参数估计量,证明了该估计量具有强相合性和渐近正态性。针对不同的可靠寿命,具体给出了使得估计效率达到最大的最优挑选抽样设计。最后,渐近相对效率和实际应用的研究结果表明:最优挑选设计的抽样效率高于简单随机抽样。     

Validity of the formal Edgeworth expansion when the underlying distribution is partly discrete

Summary Validity of the formal Edgeworth expansion for the distribution of the statistic ng(X _n /n, Y _n /n) is considered. Here X _n is a continuous variate and Y _n is a discrete variate. In general if (X _n , Y _n ) resemble the sum of i.i.d. variables and the partial derivative of g with respect to the first variable has full rank it is possible to establish an Edgeworth expansion.