Variance estimation in the central limit theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the maximum likelihood estimation (MLE) for the stationary distribution.     

基于半参数方法的模型辅助抽样估计研究

在总结现有模型辅助估计方法的基础上,本文通过构造一种半参数超总体模型,同时结合广义差分估计思想提出一种新型的模型辅助估计量。该估计量比传统的非参数和半参数回归估计利用更少、更易得到的辅助信息,即只需利用和广义回归估计相同的辅助信息,但一般会比广义回归估计拥有更高的估计精度。理论证明了该估计量是渐近设计无偏和设计一致的,其渐近设计均方误差为广义差分估计量的方差。模拟结果显示：其至少与广义回归估计一样好;对于线性程度越低的超总体模型,其估计精度比广义回归估计有越明显的提高;就本文模拟而言,光滑参数在0.04～0.12间适当取值时其会取到相对较好的估计效果。     

Asymptotic properties of kernel estimates of a regression function

The consistency of the Parzen kernel-type as well as recursive kernel estimates of a regression function is shown. The rates of the convergence are studied and compared. Moreover, the problem of selecting asymptotically optimal kernels is discussed.     

Efficiency of weighted averages

When combining estimates of a common parameter (of dimension d?1$d?1$) from independent data sets—as in stratified analyses and meta analyses—a weighted average, with weights ‘proportional’ to inverse variance matrices, is shown to have a minimal variance matrix (a standard fact when d=1$d=1$)—minimal in the sense that all convex combinations of the coordinates of the combined estimate have minimal variances. Minimum variance for the estimation of a single coordinate of the parameter can therefore be achieved by joint estimation of all coordinates using matrix weights. Moreover, if each estimate is asymptotically efficient within its own data set, then this optimally weighted average, with consistently estimated weights, is shown to be asymptotically efficient in the combined data set and avoids the need to merge the data sets and estimate the parameter in question afresh. This is so whatever additional non-common nuisance parameters may be in the models for the various data sets. A special case of this appeared in Fisher [1925. Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22, 700–725.]: Optimal weights are ‘proportional’ to information matrices, and he argued that sample information should be used as weights rather than expected information, to maintain second-order efficiency of maximum likelihood. A number of special cases have appeared in the literature; we review several of them and give additional special cases, including stratified regression analysis—proportional-hazards, logistic or linear—, combination of independent ROC curves, and meta analysis. A test for homogeneity of the parameter across the data sets is also given.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

Interaction balance for symmetrical factorial designs with generalized minimum aberration

There are two different systems of contrast parameterization when analyzing the interaction effects among the factors with more than two levels, i.e., linear-quadratic system and orthogonal components system. Based on the former system and an ANOVA model, Xu and Wu (2001) introduced the generalized wordlength pattern for general factorial designs. This paper shows that the generalized wordlength pattern exactly measures the balance pattern of interaction columns of a symmetrical design ground on the orthogonal components system, and thus an alternative angle to look at the generalized minimum aberration criterion is given. This work is partially supported by NNSF of China grant No. 10231030.     

MODEL-UNBIASEDNESS AND THE HORVITZ-THOMPSON ESTIMATOR IN FINITE POPULATION SAMPLING

Under the, notion of superpopulation models, the concept of minimum expected variance is adopted as an optimality criterion for design-unbiased estimators, i.e. unbiased under repeated sampling. In this article, it is shown that the Horvitz-Thompson estimator is optimal among such estimators if and only if it is model-unbiased, i.e. unbiased under the model. The family of linear models is considered and a sample design is suggested to preserve the model-unbiasedness (and hence the optimality) of the Horvitz-Thompson estimator. It is also shown that under these models the Horvitz-Thompson estimator together with the suggested sample design is optimal among design-unbiased estimators with any sample design (of fixed size n ) having non-zero probabilities of inclusion for all population units.     

A note on the connection between uniformity and generalized minimum aberration

Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs. This research was partially supported by the NNSF of China (No. 10441001), the Key Project of Chinese Ministry of Education (No. 105119) and the Project-sponsored by SRF for ROCS (SEM) (No. [2004]176).     

Empirical likelihood for a partially linear model with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, the empirical log-likelihood ratios are proven to be asymptotically chi-squared and the corresponding confidence regions for the parameters of interest are then constructed. The finite sample behavior of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial dataset.     

A Comparison of Survival Function Estimators in the Koziol–Green Model

In this paper, three competing survival function estimators are compared under the assumptions of the so-called Koziol– Green model, which is a simple model of informative random censoring. It is shown that the model specific estimators of Ebrahimi and Abdushukurov, Cheng, and Lin are asymptotically equivalent. Further, exact expressions for the (noncentral) moments of these estimators are given, and their biases are analytically compared with the bias of the familiar Kaplan–Meier estimator. Finally, MSE comparisons of the three estimators are given for some selected rates of censoring.     

Mean and sensitivity estimation in optional randomized response models

Optional randomized response models were introduced by Gupta et al. (2002). These models are based on the basic premise that a question may be sensitive for one respondent but may not be sensitive for another. In an optional RRT (randomized response technique) model, a respondent is asked to provide a scrambled response only if the respondent considers the question sensitive. Otherwise, the respondent provides a truthful response. The researcher does not know which type of response is provided. The proportion of respondents who provide a scrambled response is known as the sensitivity level of the question. In this paper, we estimate simultaneously the mean and the sensitivity level of a quantitative-response sensitive question using a two stage optional RRT model. The estimators are unbiased and asymptotically normally distributed. We discuss the advantages and disadvantages of using additive and multiplicative scrambling.     

Hypotheses testing: Poisson versus stress-release

We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to stationary Poisson process with known intensity against a composite one-sided parametric alternative that this is a stress-release point process. The underlying family of measures is locally asymptotically quadratic and we describe the behavior of score-function, likelihood ratio and Wald tests in the asymptotics of large samples. The results of numerical simulations are presented.     

Generalized quasi-likelihood

The quasi-score function, as defined by Wedderburn (1974) and McCullagh (1983) and so on, is a linear function of observations. The generalized quasi-score function introduced in this paper is a linear function of some unbiased basis functions, where the unbiased basis functions may be some linear functions of the observations or not, and can be easily constructed by the meaning of the parameters such as mean and median and so on. The generalized quasi-likelihood estimate obtained by such a generalized quasi-score function is consistent and has an asymptotically normal distribution. As a result, the optimum generalized quasi-score is obtained and a method to construct the optimum unbiased basis function is introduced. In order to construct the potential function, a conservative generalized estimating function is defined. By conservative, a potential function for the projected score has many properties of a log-likelihood function. Finally, some examples are given to illustrate the theoretical results. This paper is supported by NNSF project (10371059) of China and Youth Teacher Foundation of Nankai University.     

A comparison of conditional Poisson sampling versus unequal probability sampling with replacement

The variance of the Horvitz–Thompson estimator for a fixed size Conditional Poisson sampling scheme without replacement and with unequal inclusion probabilities is compared to the variance of the Hansen–Hurwitz estimator for a sampling scheme with replacement. We show, using a theorem by Gabler, that the sampling design without replacement is more efficient than the sampling design with replacement.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

Variance Estimation Based on Invariance Principles

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.     

On the estimation of the influence curve

We prove the asymptotic validity of bootstrap confidence bands for the influence curve from its usual estimator (the sensitivity curve). The proof is based on the use of Gill's generalized delta method for Hadamard differentiable operators. Some statistical applications, in particular to the estimation of asymptotic variance, are given.     

Antipodally symmetric distributions for orientation statistics

The conventional antipodally symmetric Bingham matrix distribution on the Stiefel manifold is generalized. Large sample maximum likelihood estimation and uniformity tests are discussed, and a parametric model for axial orientations (X-shapes) is suggested. A generalization of the Khatri-Mardia matrix distribution is developed to provide a model suitable for hybrids (T-shapes).     

Inferences of variance function - a parametric robust way

Tsou (2003a) proposed a parametric procedure for making robust inference for mean regression parameters in the context of generalized linear models. This robust procedure is extended to model variance heterogeneity. The normal working model is adjusted to become asymptotically robust for inference about regression parameters of the variance function for practically all continuous response variables. The connection between the novel robust variance regression model and the estimating equations approach is also provided.     

On quadratic prediction problems in finite populations

For a finite population and its linear model, Liu and Rong proposed a notion of optimal invariant quadratic unbiased prediction (OIQUP) and offered two methods for studying this notion, in which the first is incomplete. In this note, we mainly aim at fulfilling the first approach used by Liu and Rong by considering a transformed matrix equation set through permutation matrix techniques. Solvability of the matrix equation set, optimality of the resulting predictor, and equivalence of the representations of OIQUP, derived in this note and by Liu and Rong, are investigated in detail. In addition, an application to predicting population variance is conducted based on a simulated population.