Asymptotic optimality of estimators of a linear functional eeiation if the ratio of the error variances is known 2

For the problem of estimating a linear functional relation when the ratio of the error variances is known a general class of estimators is introduced. They include as special cases the instrumental variable and replication cases and some others. Conditions are given for consistency, asymptotic normality and asymptotic optimality within this class based on the variance of the limit distribution. Fisheb's lower bound for asymptotic variances is established, and under normality the asymptotically optimal estimators are shown to be best asymptotically normal. For an inhomogeneous linear relation only estimators which are invariant with respect to a translation of the origin are considered, and asymptotically optimal invariant and, under normality, best asymptotically normal invariant estimators are obtained. Several special cases are discussed.     

Effect of Violation of the Normal Assumption on MI and ML Estimators in the Analysis of Incomplete Data

Asymptotic distributions of normal-theory-based ML/MI estimators are studied in a simple regression model under general distributions with MAR missing data. The asymptotic variance of the ML/MI estimator of residuals’ variance is explicitly derived, from which it follows that the kurtosis of the error distribution primarily affects the asymptotic variance. Results of numerical simulations conducted to study finite sample properties of the estimators, conformed largely to the asymptotic results, and they also indicated interesting findings particularly for small samples, which do not follow from the asymptotic property. It is concluded that the ML estimators perform best in the situation studied here.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Inference about the Regression Parameters Using Median-Ranked Set Sampling

The ranked set samples and median ranked set samples in particular have been used extensively in the literature due to many reasons. In some situations, the experimenter may not be able to quantify or measure the response variable due to the high cost of data collection, however it may be easier to rank the subject of interest. The purpose of this article is to study the asymptotic distribution of the parameter estimators of the simple linear regression model. We show that these estimators using median ranked set sampling scheme converge in distribution to the normal distribution under weak conditions. Moreover, we derive large sample confidence intervals for the regression parameters as well as a large sample prediction interval for new observation. Also, we study the properties of these estimators for small sample setup and conduct a simulation study to investigate the behavior of the distributions of the proposed estimators.     

Optimal sample size allocation for multi-level stress testing with Weibull regression under Type-II censoring

We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.     

Immaculating the inconsistent estimator of slope parameter in measurement error model with replicated data

ABSTRACT

The measurement error model with replicated data on study as well as explanatory variables is considered. The measurement error variance associated with the explanatory variable is estimated using the complete data and the grouped data which is used for the construction of the consistent estimators of regression coefficient. These estimators are further used in constructing an almost unbiased estimator of regression coefficient. The large sample properties of these estimators are derived without assuming any distributional form of the measurement errors and the random error component under the setup of an ultrastructural model.     

Information attainable in some randomly incomplete data models

The Fisher information is intricately linked to the asymptotic (first-order) optimality of maximum likelihood estimators for parametric complete-data models. When data are missing completely at random in a multivariate setup, it is shown that information in a single observation is well-defined and it plays the same role as in the complete-data model in characterizing the first-order asymptotic optimality properties of associated maximum likelihood estimators; computational aspects are also thoroughly appraised. As an illustration, the logistic regression model with incomplete binary responses and an incomplete categorical covariate is worked out.     

Quantile Estimation in the Presence of Auxiliary Information under Negatively Associated Samples

In this article, we apply the empirical likelihood technique to propose a new class of quantile estimators in the presence of some auxiliary information under negatively associated samples. It is shown that the proposed quantile estimators are asymptotically normally distributed with smaller asymptotic variances than those of the usual quantile estimators. It is also shown that blocking technique is an useful tool in estimating asymptotic variance under negatively associated samples, which makes it possible to construct normal approximation based confidence intervals for quantiles.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Estimation of parameters of two-dimensional sinusoidal signal in heavy-tailed errors

In this paper, we consider a two-dimensional sinusoidal model observed in an additive random field. The proposed model has wide applications in statistical signal processing. The additive noise has mean zero but the variance may not be finite. We propose the least squares estimators to estimate the unknown parameters. It is observed that the least squares estimators are strongly consistent. We obtain the asymptotic distribution of the least squares estimators under the assumption that the additive errors are from a symmetric stable distribution. Some numerical experiments are performed to see how the results work for finite samples.     

Model averaging for multivariate multiple regression models

The theories and applications of model averaging have been developed comprehensively in the past two decades. In this paper, we consider model averaging for multivariate multiple regression models. In order to make use of the correlation information of the dependent variables sufficiently, we propose a model averaging method based on Mahalanobis distance which is related to the correlation of the dependent variables. We prove the asymptotic optimality of the resulting Mahalanobis Mallows model averaging (MMMA) estimators under certain assumptions. In the simulation study, we show that the proposed MMMA estimators compare favourably with model averaging estimators based on AIC and BIC weights and the Mallows model averaging estimators from the single dependent variable regression models. We further apply our method to the real data on urbanization rate and the proportion of non-agricultural population in ethnic minority areas of China.     

Relative performance of stein-rule and preliminary test estimators in linear models least squares theory

In the classical (univariare) linear model, bearing the plausibility of a subset of the regression parameters being close to a pivot, shrinkage least squares estimation of the complementary subset is considered. Based on the usual James-Stein rule, shrinkage least squares estimators are constructed, and under an asymptotic setup (allowing the shrinkage parameters to be 'close to ' the pivot), the relative performance of such estimators and the prcliminary test estimators is studied. In this context, the normality of the errors is also avoided under the same asymptotic setup. None of the shrinkage and preliminary test estimators may dominate the other (in the light of the asymptotic distributional risk criterion, as has been developed here), though each of them fares well relative to the classical least squeres estimator. The chice of the shrinkage factor is also examined properly.     

On Local Polynomial Modelling of the Additive Risk Model

The additive risk model provides an alternative modelling technique for failure time data to the proportional hazards model. In this article, we consider the additive risk model with a nonparametric risk effect. We study estimation of the risk function and its derivatives with a parametric and an unspecified baseline hazard function respectively. The resulting estimators are the local likelihood and the local score estimators. We establish the asymptotic normality of the estimators and show that both methods have the same formula for asymptotic bias but different formula for variance. It is found that, in some special cases, the local score estimator is of the same efficiency as the local likelihood estimator though it does not use the information about the baseline hazard function. Another advantage of the local score estimator is that it has a closed form and is easy to implement. Some simulation studies are conducted to evaluate and compare the performance of the two estimators. A numerical example is used for illustration.     

Extension of a Two-Stage Conditionally Unbiased Estimator of the Selected Population to the Bivariate Normal Case

A useful parameterization of the exponential failure model with imperfect signalling, under random censoring scheme, is considered to accommodate covariates. Simple sufficient conditions for the existence, uniqueness, consistency, and asymptotic normality of maximum likelihood estimators for the parameters in these models are given. The results are then applied to derive the asymptotic properties of the likelihood ratio test for a difference between failure signalling proportions between groups in a ‘one-way’ classification.     

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d_x; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models

Abstract.  This paper considers simultaneous estimation of means from several strata. A model-based approach is taken, where the covariates in the superpopulation model are subject to measurement errors. Empirical Bayes (EB) and Hierarchical Bayes estimators of the strata means are developed and asymptotic optimality of EB estimators is proved. Their performances are examined and compared with that of the sample mean in a simulation study as well as in data analysis.     

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.     

On hsu's model in regression analysis

The paper examplifies with Hsu’s model a general pattern as how to derive results of variance component estimation from well known results on mean estimation, as far as linear model theory is concerned. This ’ dispersion-mean-correspondence‘provides new and short proofs for various theorems from the literature, concerning unbiased invariant quadratic estimators with minimum BAYES risk or minimum variance. For pure variance component models, unbiased non-negative quadratic estimability is characterized in terms of the design matrices.     

A NOTE ON VARIANCE ESTIMATION FOR THE GENERALIZED REGRESSION PREDICTOR

The generalized regression (GREG) predictor is used for estimating a finite population total when the study variable is well‐related to the auxiliary variable. In 1997, Chaudhuri & Roy provided an optimal estimator for the variance of the GREG predictor within a class of non‐homogeneous quadratic estimators (H) under a certain superpopulation model M. They also found an inequality concerning the expected variances of the estimators of the variance of the GREG predictor belonging to the class H under the model M. This paper shows that the derivation of the optimal estimator and relevant inequality, presented by Chaudhuri & Roy, are incorrect.