Estimation of a linear model under microaggregation by individual ranking

Summary Microaggregation by individual ranking is one of themost commonly applied disclosure control techniques for continuous microdata. The paper studies the effect of microaggregation by individual ranking on the least squares estimation of a multiple linear regression model. It is shown that the traditional least squares estimates are asymptotically unbiased. Moreover, the least squares estimates asymptotically have the same variances as the least squares estimates based on the original (non-aggregated) data. Thus, asymptotically, microaggregation by individual ranking does not result in a loss of efficiency in the least squares estimation of a multiple linear regression model. I thank Hans Schneeweiss for very helpful discussions and comments. Financial support from the Deutsche Forschungsgemeinschaft (German Science Foundation) is gratefully acknowledged.     

Estimation in singular partitioned,reduced or transformed linear models

A singular partitioned linear model, i.e. the singular model comprising the main parameters and the nuisance parameters, can be reduced, or transformed to the form in which only linear functions concerning main parameters are involved. In the paper some properties of the best linear unbiased estimators of these functions following from these models are considered.     

Estimation of Sobol's sensitivity indices under generalized linear models

We derive explicit formulas for Sobol's sensitivity indices (SSIs) under the generalized linear models (GLMs) with independent or multivariate normal inputs. We argue that the main-effect SSIs provide a powerful tool for variable selection under GLMs with identity links under polynomial regressions. We also show via examples that the SSI-based variable selection results are similar to the ones obtained by the random forest algorithm but without the computational burden of data permutation. Finally, applying our results to the problem of gene network discovery, we identify through the SSI analysis of a public microarray dataset several novel higher-order gene–gene interactions missed out by the more standard inference methods. The relevant functions for SSI analysis derived here under GLMs with identity, log, and logit links are implemented and made available in the R package Sobol sensitivity.     

Estimation of the Sobol indices in a linear functional multidimensional model

We consider a functional linear model where the explicative variables are known stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L²([0,1])${\mathbb{L}}^2([0,1])$. We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U-statistics. We prove the asymptotic normality and the efficiency of our estimators and we compare them from a theoretical and practical point of view with classical estimators of Sobol indices.     

Estimation of linear models with anonymised panel data

The paper analyses the biasing effect of anonymising micro data by multiplicative stochastic noise on the within estimation of a linear panel model. In short panels, additional bias results from serially correlated regressors. Results in this paper are related to the project “Firms’ Panel Data and Factual Anonymisation,” which is financed by Federal Ministry of Education and Research. We would like to thank the anonymous referees for helpful comments.     

Regression analysis of informative current status data with the semiparametric linear transformation model

Many methods have been developed in the literature for regression analysis of current status data with noninformative censoring and also some approaches have been proposed for semiparametric regression analysis of current status data with informative censoring. However, the existing approaches for the latter situation are mainly on specific models such as the proportional hazards model and the additive hazard model. Corresponding to this, in this paper, we consider a general class of semiparametric linear transformation models and develop a sieve maximum likelihood estimation approach for the inference. In the method, the copula model is employed to describe the informative censoring or relationship between the failure time of interest and the censoring time, and Bernstein polynomials are used to approximate the nonparametric functions involved. The asymptotic consistency and normality of the proposed estimators are established, and an extensive simulation study is conducted and indicates that the proposed approach works well for practical situations. In addition, an illustrative example is provided.     

Consistent inference for biased sub-model of high-dimensional partially linear model

In this paper, we study a working sub-model of partially linear model determined by variable selection. Such a sub-model is more feasible and practical in application, but usually biased. As a result, the common parameter estimators are inconsistent and the corresponding confidence regions are invalid. To deal with the problems relating to the model bias, a nonparametric adjustment procedure is provided to construct a partially unbiased sub-model. It is proved that both the adjusted restricted-model estimator and the adjusted preliminary test estimator are partially consistent, which means when the samples drop into some given subspaces, the estimators are consistent. Luckily, such subspaces are large enough in a certain sense and thus such a partial consistency is close to global consistency. Furthermore, we build a valid confidence region for parameters in the sub-model by the corresponding empirical likelihood.     

Estimation of the linear EV model with censored data

In this paper, a censored linear errors-in-variables model is investigated. The asymptotic normality of the unknown parameter's estimator is obtained. Two empirical log-likelihood ratio statistics for the unknown parameter in the model are suggested. It is proved that the proposed statistics are asymptotically chi-squared under some mild conditions, and hence can be used to construct the confidence regions of the parameter of interest. Finite sample performance of the proposed method is illustrated in a simulation study.     

Dimension test approach of heteroscedasticity in the linear model

Heteroscedasticity testing has a long history and is still an important matter in the linear model. There exist many types of tests, but they are limited in use to their own specific cases and sensitive to normality. Here, we propose a dimension test approach to heteroscedasticity. The proposed test overcomes the shortcomings of the existing methods, so that it is robust to normality and is unified in sense that it is applicable in the linear model with multi-dimensional response. Numerical studies confirm that the proposed test is favorable over the existing tests with moderate sample sizes, and real data analysis is presented.     

Robust selection of variables in linear discriminant analysis

A commonly used procedure for reduction of the number of variables in linear discriminant analysis is the stepwise method for variable selection. Although often criticized, when used carefully, this method can be a useful prelude to a further analysis. The contribution of a variable to the discriminatory power of the model is usually measured by the maximum likelihood ratio criterion, referred to as Wilks’ lambda. It is well known that the Wilks’ lambda statistic is extremely sensitive to the influence of outliers. In this work a robust version of the Wilks’ lambda statistic will be constructed based on the Minimum Covariance Discriminant (MCD) estimator and its reweighed version which has a higher efficiency. Taking advantage of the availability of a fast algorithm for computing the MCD a simulation study will be done to evaluate the performance of this statistic. The presentation of material in this article does not imply the expression of any opinion whatsoever on the part of Austro Control GmbH and is the sole responsibility of the authors.     

Estimation in a model for a semi-Markov process with covariates under right-censoring

A semi-Markov model with covariates is proposed for a multi-state process with a finite number of states such that the transition probabilities between the states and the distribution functions of the duration times between the occurrence of two states depend on a discrete covariate. The hazard rates for the time elapsed between two successive states depend on the covariate through a proportional hazards model involving a set of regression parameters, while the transition probabilities depend on the covariate in an unspecified way. We propose estimators for these parameters and for the cumulative hazard functions of the sojourn times. A difficulty comes from the fact that when a sojourn time in a state is right-censored, the next state is unknown. We prove that our estimators are consistent and asymptotically Gaussian under the model constraints.     

Computations of predictors/estimators under a linear random-effects model with parameter restrictions

This paper considers the problem of simultaneously predicting/estimating unknown parameter spaces in a linear random-effects model with both parameter restrictions and missing observations. We shall establish explicit formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameters in such a model, and derive a variety of mathematical and statistical properties of the BLUPs under general assumptions. We also discuss some matrix expressions related to the covariance matrix of the BLUP, and present various necessary and sufficient conditions for several equalities and inequalities of the covariance matrix of the BLUP to hold.     

Stein-rule estimator under inclusion of superfluous variables in linear regression models

Stein-rule estimation is a well-known method to improve the unbiased OLSE in the sense of smaller Mean-Square-Error. The paper is investigating the behaviour of this efficiency relation in case of misspecification of the linear model caused by inclusion of superfluous variables     

Estimation in a linear regression model under the Kullback–Leibler loss and its application to model selection

This paper is concerned with the problem of constructing a good predictive distribution relative to the Kullback–Leibler information in a linear regression model. The problem is equivalent to the simultaneous estimation of regression coefficients and error variance in terms of a complicated risk, which yields a new challenging issue in a decision-theoretic framework. An estimator of the variance is incorporated here into a loss for estimating the regression coefficients. Several estimators of the variance and of the regression coefficients are proposed and shown to improve on usual benchmark estimators both analytically and numerically. Finally, the prediction problem of a distribution is noted to be related to an information criterion for model selection like the Akaike information criterion (AIC). Thus, several AIC variants are obtained based on proposed and improved estimators and are compared numerically with AIC as model selection procedures.     

Estimation of a finite population distribution function based on a linear model with unknown heteroscedastic errors

The authors consider a finite population ρ = {(Y_k, x_k), k = 1,…,N} conforming to a linear superpopulation model with unknown heteroscedastic errors, the variances of which are values of a smooth enough function of the auxiliary variable X for their nonparametric estimation. They describe a method of the Chambers‐Dunstan type for estimation of the distribution of {Y_k, k = 1,…, N} from a sample drawn from without replacement, and determine the asymptotic distribution of its estimation error. They also consider estimation of its mean squared error in particular cases, evaluating both the analytical estimator derived by “plugging‐in” the asymptotic variance, and a bootstrap approach that is also applicable to estimation of parameters other than mean squared error. These proposed methods are compared with some common competitors in simulation studies.     

Improved Liu estimator in a linear regression model

In this paper, we mainly aim to introduce the notion of improved Liu estimator (ILE) in the linear regression model y=Xβ+e. The selection of the biasing parameters is investigated under the PRESS criterion and the optimal selection is successfully derived. We make a simulation study to show the performance of ILE compared to the ordinary least squares estimator and the Liu estimator. Finally, the main results are applied to the Hald data.     

Confidence sets in a linear regression model for interval data

The construction of confidence sets for the parameters of a flexible simple linear regression model for interval-valued random sets is addressed. For that purpose, the asymptotic distribution of the least-squares estimators is analyzed. A simulation study is conducted to investigate the performance of those confidence sets. In particular, the empirical coverages are examined for various interval linear models. The applicability of the procedure is illustrated by means of a real-life case study.     

Small area estimation under a multivariate linear model for repeated measures data

In this article, small area estimation under a multivariate linear model for repeated measures data is considered. The proposed model aims to get a model which borrows strength both across small areas and over time. The model accounts for repeated surveys, grouped response units, and random effects variations. Estimation of model parameters is discussed within a likelihood based approach. Prediction of random effects, small area means across time points, and per group units are derived. A parametric bootstrap method is proposed for estimating the mean squared error of the predicted small area means. Results are supported by a simulation study.     

A bayesian analysis of the mixed linear model

The mixed model is defined. The exact posterior distribution for the fixed effect vector is obtained. The exact posterior distribution for the error variance is obtained. The exact posterior mean and variance of a Bayesian estimator for the variances of random effects is also derived. All computations are non-iterative and avoid numerical integrations.     

Nonparametric estimation of linear functionals of a bivariate distribution under univariate censoring

In the presence of univariate censoring, a class of nonparametric estimators is proposed for linear functionals of a bivariate distribution of paired failure times. The estimators are shown to be root-n consistent and asymptotically normal. An adjusted empirical log-likelihood ratio statistic is developed and proved to follow a chi-square distribution asymptotically. Two types of confidence intervals, based on the normal approximation method and the empirical likelihood method, respectively, are constructed to make inference about the linear functionals. Their performance is evaluated in several simulation studies and a real example.