Linearisation for Variance Estimation by Means of Sampling Indicators: Application to Non‐response

In the presence of non‐response, it is usual to impute missing values, to reweight responding units and to adapt the estimators accordingly. The computation of the precision of the estimators becomes rapidly complex; it must take into account the sampling design, the treatment and the refinement of the estimators. In the absence of non‐response, it is possible to linearise estimators with respect to the sampling indicators to compute explicit variance estimators. In this paper, we extend this linearisation method to deal with non‐response. It becomes particularly straightforward to compute explicit variance estimators. Some known results are revisited in a simpler way than the usual developments, and new results for complex estimators are proposed. A simulation study evaluates the proposed methodology.     

Restricted maximum likelihood estimation under Eisenhart model Ill

For a balanced two-way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non-negativity requirements of the variance components by L ee and K apadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non-negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

On estimation of variance components*

In this survey paper the estimation of variance components is given. The least squares approach in variance component estimation is a unifying principle which includes the analysis of variance estimators and the MINQUE. When normality is assumed the maximum likelihood estimators can be used. Many variance component estimators are not permissible because they are not non-negative. The development of non-negative variance component estimators is indicated.     

Preliminary test estimators of the parameters of simple linear model with measurement error

In this paper, we discuss the properties of preliminary test estimators (PTE) of the parameters of simple linear model with measurement error (ME model) when the slope of the linear model is suspected to be zero. Expressions of the bias, MSE and efficiencies are obtained under conditional as well as unconditional situations with known reliability coefficient. Conditional model results are compared to the standard model without measurement error. We also provide the unconditional model analysis in finite samples. Asymptotic theory under local alternatives is developed when the variance of measurement error or the ratio of the variance of the model error relative to the variance of the measurement error is known. Asymptotic expressions of bias and MSE of the estimators along with their efficiencies are obtained. In every case, it is shown that the measurement error tend to increase the variability of the estimators compared to the estimators without measurement error. Graphs and tables are provided to see these results and to determine optimum level of significance for minimum guaranteed efficiency. Received October 2001 RID="*" ID="*"  A. K. Md. E. Saleh is a Distinguished Research Professor and H. M. Kim is a Ph.D. candidate in the School of Mathematics and Statistics, Carleton University, Ottawa. Acknowledgment. The authors gratefully acknowledge the constructive suggestion of the referees to improve the paper. The research is supported by NSERC grant A3088.     

Estimation procedures for a family of density functions representing various life-testing models

A family of density functions is considered which contains several life-testing models as specific cases. Uniformly minimum variance unbiased estimators are obtained for the positive and negative powers of the parameter, moments and reliability function. These general results provide the estimators for the specific models.     

Panel data models with spatially correlated error components

In this paper we consider a panel data model with error components that are both spatially and time-wise correlated. The model blends specifications typically considered in the spatial literature with those considered in the error components literature. We introduce generalizations of the generalized moments estimators suggested in Kelejian and Prucha (1999. A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509–533) for estimating the spatial autoregressive parameter and the variance components of the disturbance process. We then use those estimators to define a feasible generalized least squares procedure for the regression parameters. We give formal large sample results for the proposed estimators. We emphasize that our estimators remain computationally feasible even in large samples.     

Two kinds of variance/covariance estimates in linear mixed models

For longitudinal data, the within-subject covariance matrix plays an important role in statistical inference and it is of great interest to investigate this. In the paper, two kinds of estimators are investigated for the random effect covariance matrix D ₁ and the error variance σ ² in linear mixed models. One is to estimate D ₁ first and then to estimate σ ²; the other kind is to estimate σ ² first and then for D ₁. Both kinds of estimators are consistent. The covariance matrices of these covariance estimators and the variances of these two error variance estimators are calculated. In particular, the mean square errors of these estimators are also derived for one dimensional random effects. Besides, a simulation study is conducted to investigate the performances of these estimators.     

On estimating the mean of the selected normal population under the LINEX loss function

Following Parsian and Farsipour (1999), we consider the problem of estimating the mean of the selected normal population, from two normal populations with unknown means and common known variance, under the LINEX loss function. Some admissibility results for a subclass of equivariant estimators are derived and a sufficient condition for the inadmissibility of an arbitrary equivariant estimator is provided. As a consequence, several of the estimators proposed by Parsian and Farsipour (1999) are shown to be inadmissible and better estimators are obtained. Received January 2001/Revised May 2002     

Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

Estimating regression coefficients from survey data by asymptotic design-cum-model based approach

Postulating a linear regression of a variable of interest on an auxiliary variable with values of the latter known for all units of a survey population, we consider appropriate ways of choosing a sample and estimating the regression parameters. Recalling Thomsen’s (1978) results on non-existence of ‘design-cum-model’ based minimum variance unbiased estimators of regression coefficients we apply Brewer’s (1979) ‘asymptotic’ analysis to derive ‘asymptotic-design-cummodel’ based optimal estimators assuming large population and sample sizes. A variance estimation procedure is also proposed.     

Stratification,midzuno-sen strategy and nonnegative unbiased variance estimation

When the ratio method is appropriate for estimating the population total one is faced with the problem of nonavailability of uniformly nonnegative unbiased variance estimators (nnuve). Here we highlight the twofold role of stratification in that it not only improves the efficiency of the ratio method of estimation but it also enhances the chances of getting uniformly nonnegative unbiased variance estimators.     

基于EMB多重插补法的线性模型系数估计量的模拟研究

基于EMB多重插补法的线性模型系数估计量,分析其统计性质,并与PMM多重插补法以及DA插补法进行比较。模拟结果显示,随着无回答率增加,系数估计量的偏差绝对值、均方误差呈递增趋势,估计方差的递增趋势相对更显著。在完全随机无回答机制或随机无回答机制下,建议插补重数为15。在依赖被解释变量的非随机无回答机制下,建议插补重数可适当增大。在依赖其他变量的非随机无回答机制下,估计量的均方误差和估计方差的差异大,使用EMB多重插补法要谨慎。     

Market microstructure noise,integrated variance estimators,and the accuracy of asymptotic approximations

A growing literature has been advocating consistent kernel estimation of integrated variance in the presence of financial market microstructure noise. We find that, for realistic sample sizes encountered in practice, the asymptotic results derived for the proposed estimators may provide unsatisfactory representations of their finite sample properties. In addition, the existing asymptotic results might not offer sufficient guidance for practical implementations. We show how to optimize the finite sample properties of kernel-based integrated variance estimators. Empirically, we find that their suboptimal implementation can, in some cases, lead to little or no finite sample gains when compared to the classical realized variance estimator. Significant statistical and economic gains can, however, be recovered by using our proposed finite sample methods.     

Nonlinear regression models with single‐index heteroscedasticity

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

The structure of simultaneous equations estimators

The formula for the Full Information Maximum Likelihood Estimator for a linear simultaneous system (with finite variance, serially independent errors) is demonstrated to be an estimator generating equation for econometrics in that all presently known estimators are readily derivable from that formula if they are considered as numerical approximations to its solution. Further, the approach immediately classifies the resulting estimators into asymptotically equivalent groups. The method is then generalised to encompass the large class of estimators for dynamic systems with (vector) autoregressive errors. The very close relationship between estimation rules and non-linear optimisation algorithms is highlighted.     

The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.     

On the optimum number of levels in the one-way model with random effects

The optimal number of levels is studied for the one-way random model with normally distributed effects. The optimum criteria used are based on the variances of the traditional analysis of variance estimators of the variance components. Exact solutions are compared to earlier results based on lower bounds of the sampling variances. Comparisons are also made to the large-sample variances of the estimates based on restricted maximum likelihood. Received February 2002     

Threshold estimation of Markov models with jumps and interest rate modeling

We reconstruct the level-dependent diffusion coefficient of a univariate semimartingale with jumps which is observed discretely. The consistency and asymptotic normality of our estimator are provided in the presence of both finite and infinite activity (finite variation) jumps. Our results rely on kernel estimation, using the properties of the local time of the data generating process, and the fact that it is possible to disentangle the discontinuous part of the state variable through those squared increments between observations not exceeding a suitable threshold function. We also reconstruct the drift and the jump intensity coefficients when they are level-dependent and jumps have finite activity, through consistent and asymptotically normal estimators. Simulated experiments show that the newly proposed estimators perform better in finite samples than alternative estimators, and this allows us to reexamine the estimation of a univariate model for the short term interest rate, for which we find fewer jumps and more variance due to the diffusion part than previous studies.