Second-order nonlinear least squares estimation

The ordinary least squares estimation is based on minimization of the squared distance of the response variable to its conditional mean given the predictor variable. We extend this method by including in the criterion function the distance of the squared response variable to its second conditional moment. It is shown that this “second-order” least squares estimator is asymptotically more efficient than the ordinary least squares estimator if the third moment of the random error is nonzero, and both estimators have the same asymptotic covariance matrix if the error distribution is symmetric. Simulation studies show that the variance reduction of the new estimator can be as high as 50% for sample sizes lower than 100. As a by-product, the joint asymptotic covariance matrix of the ordinary least squares estimators for the regression parameter and for the random error variance is also derived, which is only available in the literature for very special cases, e.g. that random error has a normal distribution. The results apply to both linear and nonlinear regression models, where the random error distributions are not necessarily known.     

Consistent estimation of coefficients in measurement error models with replicated observations

For the estimation of coefficients in a measurement error model, the least squares method utilizing original observations and averaged observations over replications provides inconsistent estimators. Based on these, consistent estimators are formulated and asymptotic properties are analyzed.     

Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

Asymptotics of Estimating Equations under Natural Conditions

In a variety of statistical problems one needs to solve an equation in order to get an estimator. We consider the large sample properties of such estimators generated from samples that are not necessarily identically distributed. Very general assumptions that lead to the existence, strong consistency, and asymptotic normality of the estimators are given. A number of results that are useful in verifying the general assumptions are given and an example illustrates their use. General applications to maximum likelihood, iteratively reweighted least squares, and robust estimation are discussed briefly.     

Bridge estimators and the adaptive lasso under heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense of Fan and Li (2001) if the oracle is based on weighted least squares. In order to address this problem we introduce weighted penalized least squares methods and demonstrate their advantages by asymptotic theory and by means of a simulation study.     

Asymptotic results in segmented multiple regression

This paper studies the asymptotic behavior of the least squares estimators in segmented multiple regression. For a model with more than one partitioning variable, each of which has one or more change-points, we study the asymptotic properties of the estimated change-points and regression coefficients. Using techniques in empirical process theory, we prove the consistency of the least squares estimators and also establish the asymptotic normality of the estimated regression coefficients. For the estimated change-points, we obtain their consistency at the rates of or 1/n, with or without continuity constraints, respectively. The change-points estimated under the continuity constraints are also shown to asymptotically have a multivariate normal distribution. For the case where the regression mean functions are not assumed to be continuous at the change-points, the asymptotic distribution of the estimated change-points involves a step function process, whose distribution does not follow a well-known distribution.     

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

A generalized maximum entropy estimator to simple linear measurement error model with a composite indicator

We extend the simple linear measurement error model through the inclusion of a composite indicator by using the generalized maximum entropy estimator. A Monte Carlo simulation study is proposed for comparing the performances of the proposed estimator to his counterpart the ordinary least squares “Adjusted for attenuation”. The two estimators are compared in term of correlation with the true latent variable, standard error and root mean of squared error. Two illustrative case studies are reported in order to discuss the results obtained on the real data set, and relate them to the conclusions drawn via simulation study.     

系数为梯形模糊数的模糊回归分析的最小二乘法

由于模糊数往往可以用梯形模糊数来逼近,因此对梯形模糊数的模糊回归模型的研究就有一定的实用价值.采用最小二乘的方法,针对输入为精确数、输出和回归系数都是梯形模糊数的模糊线性回归模型,讨论了该模型回归系数的最小二乘估计及误差项的估计,实例说明了提出的参数估计的拟合度比较好.     

Coefficients of determination for least absolute deviation analysis

The least-absolute deviation or l₁ analysis of a linear model is an important alternative to the classical least squares analysis from the point of view of efficiency for longer-tailed error distributions and robustness to the presence of outliers. In this paper two coefficients of determination are proposed for the least-absolute deviation analysis. It is shown that they have desirable properties as measures of multiple association. Both fixed and random predictor variable cases are considered. In the case of random predictor variables, the sample coefficients of determination are shown to be consistent estimators of appropriate population parameters.     

Asymptotic properties of the least squares estimators of the parameters of the chirp signals

Chirp signals are quite common in different areas of science and engineering. In this paper we consider the asymptotic properties of the least squares estimators of the parameters of the chirp signals. We obtain the consistency property of the least squares estimators and also obtain the asymptotic distribution under the assumptions that the errors are independent and identically distributed. We also consider the generalized chirp signals and obtain the asymptotic properties of the least squares estimators of the unknown parameters. Finally we perform some simulations experiments to see how the asymptotic results behave for small sample and the performances are quite satisfactory.     

Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition

In this paper, we consider the statistical inference for the partially liner varying coefficient model with measurement error in the nonparametric part when some prior information about the parametric part is available. The prior information is expressed in the form of exact linear restrictions. Two types of local bias-corrected restricted profile least squares estimators of the parametric component and nonparametric component are conducted, and their asymptotic properties are also studied under some regularity conditions. Moreover, we compare the efficiency of the two kinds of parameter estimators under the criterion of Lo?ner ordering. Finally, we develop a linear hypothesis test for the parametric component. Some simulation studies are conducted to examine the finite sample performance for the proposed method. A real dataset is analyzed for illustration.     

Error Process Indexed by Bandwidth Matrices in Multivariate Local Linear Smoothing

We focus on nonparametric multivariate regression function estimation by locally weighted least squares. The asymptotic behavior for a sequence of error processes indexed by bandwidth matrices is derived. We discuss feasible data-driven consistent estimators minimizing asymptotic mean squared error or efficient estimators reducing asymptotic bias at points where opposite sign curvatures of the regression function are present in different directions.     

A rough set approach to intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems.     

Small-sample properties of ML,COLS, and DEA estimators of frontier models in the presence of heteroscedasticity

The purpose of this paper is to examine the small sample properties of maximum likelihood (ML), corrected ordinary least squares (COLS), and data envelopment analysis (DEA) estimators of the parameters in frontier models in the presence of heteroscedasticity in the two-sided, or measurement, error term. Using Monte Carlo methods, we find that heteroscedasticity in the two-sided error term introduces substantial biases into ML, COLS, and DEA estimators. Although none of the estimators perform well, both ML and COLS are found to be superior to DEA in the presence of heteroscedasticity in the two-sided error.     

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This paper concerns with the estimation of a fixed effects panel data partially linear regression model with the idiosyncratic errors being an autoregressive process. For fixed effects short time series panel data, the commonly used autoregressive error structure fitting method will not result in a consistent estimator of the autoregressive coefficients. Here we propose an alternative estimation and show that the resulting estimator of the autoregressive coefficients is consistent and this method is workable for any order autoregressive error structure. Moreover, combining the B-spline approximation, profile least squares dummy variable (PLSDV) technique and consistently estimated the autoregressive error structure, we develop a weighted PLSDV estimator for the parametric component and a weighted B-spline series (BS) estimator for the nonparametric component. The weighted PLSDV estimator is shown to be asymptotically normal and more asymptotically efficient than the one which ignores the error autoregressive structure. In addition, this paper derives the asymptotic bias of the weighted BS estimator and establish its asymptotic normality as well. Simulation studies and an example of application are conducted to illustrate the finite sample performance of the proposed procedures.     

A Two-Step Smoothing Method for Varying-Coefficient Models with Repeated Measurements

Datasets involving repeated measurements over time are common in medical trials and epidemiological cohort studies. The outcomes and covariates are usually observed from randomly selected subjects, each at a set of possibly unequally spaced time design points. One useful approach for evaluating the effects of covariates is to consider linear models at a specific time, but the coefficients are smooth curves over time. We show that kernel estimators of the coefficients that are based on ordinary local least squares may be subject to large biases when the covariates are time-dependent. As a modification, we propose a two-step kernel method that first centers the covariates and then estimates the curves based on some local least squares criteria and the centered covariates. The practical superiority of the two-step kernel method over the ordinary least squares kernel method is shown through a fetal growth study and simulations. Theoretical properties of both the two-step and ordinary least squares kernel estimators are developed through their large sample mean squared risks.     

A fundamental relation between predictor identification and power spectrum estimation

Summary An asymptotic linear relation between the FPE (final prediction error) of the predictor obtained by using the least squares estimate of autoreggression coefficients and the integrated relative mean square error of the estimate of power spectrum obtained from the fitted autoregression model is established. This relation provides a link between the predictor identification and the power spectrum estimation and has already been used as a theoretical background of a former paper [3]. The Institute of Statistical Mathematics