Multinomial distributions applied to random sampling of particulate materials

When sampling a batch consisting of particulate material, the distribution of a sample estimator can be characterized using knowledge about the sample drawing process. With Bernoulli sampling, the number of particles in the sample is binomially distributed. Because this is rarely realized in practice, we propose a sampling design in which the possible samples have a nearly equal mass. Expected values and variances of the sample estimator are calculated. It is shown that the sample estimator becomes identical to the Horvitz–Thompson estimator in the case of a large batch-to-sample mass ratio and a large sample mass. Simulations and experiments were performed to test the theory. Simulations confirm that the round-off error due to the discrete nature of particles is negligible for large sample sizes. Sampling experiments were carried out with a mixture of PolyPropylene (PP) and PolyTetraFluorEthylene (PTFE) spheres suspended in a viscous medium. The measured and theoretical variations are in good agreement.     

An optimum sampling design to base HT-method of estimating a finite population total

The problem of constructing a sampling design with the value of the sum of second order inclusion probabilities attaining its lower bound for non-integral values of the expected effective size of a sample in the design has been considered in this paper. If the values of the characteristic of interest on all the units in the population are non-negative the design is admissible (in the sense of variance) with respect to Horvitz-Thompson estimator in the class of designs with the same set of values of the first order inclusion probabilities of the units. Again such a design is best to use Horvitz-Thompson estimator of population total in the sense of smallest average variance of the estimator under a special superpopulatio model.     

On two properties of an unequal probability sampling scheme

Summary For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.     

Comparison of sampling strategies

Summary Horvitz andThompson [1952] considered varying probability sampling method in general and furnished an unbiased estimator of the population total.Rao, Hartley andCochran [1962] proposed a simple procedure of unequal probability sampling with replacement. It leads to an estimator of the population total having smaller variance than is obtained by sampling with replacement. An attempt has been made in the present paper to compare efficiencies ofHorvitz-Thompson's estimator with that due toRao, hartley andCochran. It is demonstrated that the generalized ps sampling strategy consisting of the design with _i , the probability of inclusion of thei-th population unit in the sample proportional to the modified size together withHorvitz-Thompson's estimator is superior toRao, Hartley andCochran's sampling strategy under a general super-population model.     

Sequential estimation of normal mean under asymmetric loss function with a shrinkage stopping rule

The problem of estimating a normal mean with unknown variance is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, a sequential stopping rule and two sequential estimators of the mean are proposed and second-order asymptotic expansions of their risk functions are derived. It is demonstrated that the sample mean becomes asymptotically inadmissible, being dominated by a shrinkage-type estimator. Also a shrinkage factor is incorporated in the stopping rule and similar inadmissibility results are established. Received September 1997     

Mixed random systematic sampling designs

In this paper, an alternative sampling procedure that is a mixture of simple random sampling and systematic sampling is proposed. It results in uniform inclusion probabilities for all individual units and positive inclusion probabilities for all pairs of units. As a result, the proposed sampling procedure enables us to estimate the population mean unbiasedly using the ordinary sample mean, and to provide an unbiased estimator of its sampling variance. It is also found that the suggested sampling procedure performs well especially when the size of simple random sample is small. Received August 2001     

On the backfitting algorithm for additive regression models

We analyse additive regression model fitting via the backfitting algorithm. We show that in the case of a large class of curve estimators, which includes regressograms, simple step-by-step formulae can be given for the back-fitting algorithm. The result of each cycle of the algorithm may be represented succinctly in terms of a sequence of d projections in n-dimensional space, where d is the number of design coordinates and n is sample size. It follows from our formulae that the limit of the algorithm is simply the projection of the data onto that vector space which is orthogonal to the space of all n-vectors fixed by each of the projections. The formulae also provide the convergence rate of the algorithm, the variance of the backfitting estimator, consistency of the estimator, and the relationship of the estimator to that obtained by directly minimizing mean squared distance.     

Repeated substitution method: The ratio estimator for the population variance

This paper proposes a new unbiased estimator for the population variance in finite population sample surveys using auxiliary information. This estimator has a smaller mean squared error than the conventional unbiased estimator, the ratio estimator established by Isaki (1983) and it has the same precision than the regression estimator. Furthermore, it is a much more interesting estimator from the computation viewpoint.     

人口普查净误差估计中的双系统估计量研究

评估普查计数的完整性已经成为五年或十年一次的人口普查不可分割的一部分。评估通常采取质量评估调查的方式。该调查建立在双系统估计量基础上。考虑到普查登记质量及人口移动因素,这三份人口登记名单可提供7个总体指标。由于质量评估调查采取抽样的方式进行,所以这些总体指标要用样本来构造其估计量。基于复杂抽样方法形成的双系统估计量没有现成的方差公式计算其方差。通常使用分层刀切方差估计量近似计算双系统估计量的方差。这需要计算第一步所有样本抽样单位的复制权数。本文将对双系统估计量构造的各个环节进行理论与实践相结合的阐述,深入解析其中深层次的理论问题,为基础理论研究做出贡献,另外也将探讨基于双系统估计量的合成估计量在区域人口数目估计中的应用问题。     

M-estimation in the presence of unequal scale

Abstract  The class of weighted M-estimators is defined. The ratio of the asymptotic variance of the weighted estimator to the asymptotic variance of the optimally weighted estimator is defined as the inefficiency. A K antorovich inequality is proved, its implications are investigated for the misweighted mean and misweighted median, and the results are applied to a batch of demographic data.     

Effective nonparametric estimation in the case of severely discretized data

Often economic data are discretized or rounded to some extent. This paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel-based estimators to behave poorly. The estimator proposed here is a weighted average of neighboring frequency estimators, and the weights are composed of cubic B-splines. Interestingly, we show that this estimator can have both a smaller bias and variance than frequency estimators. As a means to obtain asymptotic normality and rates of convergence, we assume that the discreteness becomes finer as the sample size increases.     

Indirect inference for dynamic panel models

Maximum likelihood (ML) estimation of the autoregressive parameter of a dynamic panel data model with fixed effects is inconsistent under fixed time series sample size and large cross section sample size asymptotics. This paper proposes a general, computationally inexpensive method of bias reduction that is based on indirect inference, shows unbiasedness and analyzes efficiency. Monte Carlo studies show that our procedure achieves substantial bias reductions with only mild increases in variance, thereby substantially reducing root mean square errors. The method is compared with certain consistent estimators and is shown to have superior finite sample properties to the generalized method of moment (GMM) and the bias-corrected ML estimator.     

On the inefficiency of the restricted maximum likelihood

The restricted maximum likelihood is preferred by many to the full maximum likelihood for estimation with variance component and other random coefficient models, because the variance estimator is unbiased. It is shown that this unbiasedness is accompanied in some balanced designs by an inflation of the mean squared error. An estimator of the cluster‐level variance that is uniformly more efficient than the full maximum likelihood is derived. Estimators of the variance ratio are also studied.     

Estimation in Surveys Using Conditional Inclusion Probabilities: Simple Random Sampling

In survey sampling, auxiliary information on the population is often available. The aim of this paper is to develop a method which allows one to take into account such auxiliary information at the estimation stage by means of conditional bias adjustment. The basic idea is to attempt to construct a conditionally unbiased estimator. Four estimators that have a small conditional bias with respect to a statistic are proposed. It is shown that many of the estimators used in the literature in the case of simple random sampling can be obtained by using this estimation principle. The problem of simple random sampling with replacement, poststratification, and adjustment of a 2 x 2 dimensional contingency table to marginal totals are discussed in the conditional framework. Finally it is shown that the regression estimator can be viewed as an approximation of an application of the conditional principle.     

Estimating the variance of the ratio estimator for theMidzuno-Sen sampling scheme

Summary The problem of estimating the variance of the ratio estimator for theMidzuno-Sen sampling scheme is further studied in this paper. Sufficient conditions are derived for which the suggested variance estimator is always positive definite.     

On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.     

Improved Neymanian analysis for 2K factorial designs with binary outcomes

2^K factorial designs are widely adopted by statisticians and the broader scientific community. In this short note, under the potential outcomes framework, we adopt the partial identification approach and derive the sharp lower bound of the sampling variance of the estimated factorial effects, which leads to an “improved” Neymanian variance estimator that mitigates the overestimation issue suffered by the classic Neymanian variance estimator.     

Sampling with probabilities proportional to the variable of interest

To estimate the mean sojourn time, a sample of Tilburg fair visitors was asked for the duration of their stay on the fair grounds. The longer a visitor's sojourn, the larger his/her probability of being interviewed will be; therefore, longer sojourn times will be overrepresented in the sample. As a consequence, the arithmetic sample mean is not a good estimator.
The paper places this problem against a theoretical background. Sampling with unequal probabilities is considered in a general context. The special case that the sampling probabilities are a function of the variable under investigation, is discussed in detail. As a better estimator the harmonic mean of the observations is presented. Most properties of this estimator are difficult to derive analytically, but a suitable variance estimator is derived. The behavior of estimator and variance estimator is studied in a number of quite different examples.     

Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

On efficiency of the ratio estimator

Summary The exact mean square error for the ratio estimator of a finite population total based on simple random sampling without replacement is shown to have an expected value less than that of the variance of the ratio estimator based on Midzuno’s scheme, under a usual super-population model.