Error Matrix Sense Research Articles

Revisiting State Estimation for Linear Systems with State and/or Filter Equality Constraints*

This article revisits optimal state estimation, in the mean squared error matrix sense, for linear systems with state and/or filter subject to linear equality constraints (LECs). First, it is shown that the conventional Wiener filter (WF) form incorporates any LECs on the state, thus yielding a filter subject to the same LECs. Conversely, an optimal linear filter subject to LECs (or linear equality gain constraints) in general does not exist. Therefore, adding LECs on the WF or WF gain matrix either leaves unchanged or degrades the constrained WF performance w.r.t. the unconstrained WF. Since the Kalman filter (KF) and Kalman predictor (KP) are recursive WF forms for linear discrete state-space (LDSS) systems, the same results hold for both estimators, which is in contradiction with several existing results in the literature. Actually, even if these existing results are mathematically correct, however, they have been derived for unsuitable assumed LDSS models where the state is surprisingly not compliant with the assumed LECs. Indeed, it is shown that for suitable assumed state models, both standard KF and KP forms satisfy the assumed LECs, making any additional projection step superfluous.

Performance of stochastic restricted and unrestricted two-parameter estimators in linear mixed models

In this article, two parameter estimation using penalized likelihood method in the linear mixed model is proposed. In addition, by considering the stochastic linear restriction for the vector of fixed effects parameters we are introduced the stochastic restricted two parameter estimation. Methods are proposed for estimating variance parameters when unknown. Also, the superiority conditions of the two parameter estimator over the best linear unbiased estimator, and the stochastic restricted two parameter estimator over the stochastic restricted best linear unbiased estimator are obtained under the mean square error matrix sense. Methods are proposed for estimating of the biasing parameters. Finally, a simulation study and a numerical example are given to evaluate the proposed estimators

Optimal stochastic restricted logistic estimator

It is well known that the use of prior information in the logistic regression improves the estimates of regression coefficients when multicollinearity presents. This prior information may be in the form of exact or stochastic linear restrictions. In this article, in the presence of stochastic linear restrictions, we propose a new efficient estimator, named Stochastic restricted optimal logistic estimator for the parameters in the logistic regression models when the multicollinearity presents. Further, conditions for the superiority of the new optimal estimator over some existing estimators are derived with respect to the mean square error matrix sense. Moreover, a Monte Carlo simulation study and a real data example are provided to illustrate the performance of the proposed optimal estimator in the scalar mean square error sense.

On the Stochastic Restricted Modified Almost Unbiased Liu Estimator in Linear Regression Model

In this article, we propose a new biased estimator, namely stochastic restricted modified almost unbiased Liu estimator by combining modified almost unbiased Liu estimator (MAULE) and mixed estimator (ME) when the stochastic restrictions are available and the multicollinearity presents. The conditions of superiority of the proposed estimator over the ordinary least square estimator, ME, ridge estimator, Liu estimator, almost unbiased Liu estimator, stochastic restricted Liu estimator and MAULE in the mean squared error matrix sense are obtained. Finally, a numerical example and a Monte Carlo simulation are given to illustrate the theoretical findings.

A New Stochastic Restricted Liu Estimator for the Logistic Regression Model

In order to overcome the well-known multicollinearity problem, we propose a new Stochastic Restricted Liu Estimator in logistic regression model. In the mean square error matrix sense, the new estimation is compared with the Maximum Likelihood Estimation, Liu Estimator Stochastic Restricted Maximum Likelihood Estimator etc. Finally, a numerical example and a Monte Carlo simulation are given to explain some of the theoretical results.

A new stochastic restricted Liu-type estimator in linear regression model

ABSTRACTIn this paper, we proposed an alternative stochastic restricted Liu-type estimator for the vector of parameters in a linear regression model. The new estimator is a combination of ordinary mixed estimator and Liu-Type estimator , which was proposed by Liu. Necessary and sufficient conditions for the superiority of new stochastic restricted Liu-type estimator over the ordinary mixed estimator and the Liu-type estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct and are not correct. In particular, we showed that was superior in the mean squared error matrix sense over both and to the Liu-type estimator introduced by Liu (2003).

Modified Almost Unbiased Liu Estimator in Linear Regression Model

In this paper, we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator (AULE) and ridge estimator (RE) in a linear regression model when multicollinearity presents among the independent variables. Necessary and sufficient conditions for the proposed estimator over the ordinary least square estimator, RE, AULE and Liu estimator (LE) in the mean squared error matrix sense are derived, and the optimal biasing parameters are obtained. To illustrate the theoretical findings, a Monte Carlo simulation study is carried out and a numerical example is used.

A stochastic restricted k–d class estimator

In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakallıoğlu and S. Kaçiranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.

A new ridge-type estimator in stochastic restricted linear regression

In this paper, we propose a new ridge-type estimator called the weighted mixed ridge estimator by unifying the sample and prior information in linear model with additional stochastic linear restrictions. The new estimator is a generalization of the weighted mixed estimator [B. Schaffrin and H. Toutenburg, Weighted mixed regression, Zeitschrift fur Angewandte Mathematik und Mechanik 70 (1990), pp. 735–738] and ordinary ridge estimator (ORE) [A.E. Hoerl and R.W. Kennard, Ridge regression: Biased estimation for non-orthogonal problems, Technometrics 12 (1970), pp. 55–67]. The performances of this new estimator against the weighted mixed estimator, ORE and the mixed ridge estimator [Y.L. Li and H. Yang, A new stochastic mixed ridge estimator in linear regression, Stat. Pap. (2008) (in press, DOI 10.1007/s00362-008-0169-5)] are examined in terms of the mean squared error matrix sense. Finally, a numerical example and a Monte Carlo simulation are also given to show the theoretical results.

Two kinds of restricted modified estimators in linear regression model

In this paper, we introduce two kinds of new restricted estimators called restricted modified Liu estimator and restricted modified ridge estimator based on prior information for the vector of parameters in a linear regression model with linear restrictions. Furthermore, the performance of the proposed estimators in mean squares error matrix sense is derived and compared. Finally, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.

A new Liu-type estimator in linear regression model

In this paper, we introduce a new Liu-type estimator called modified Liu estimator based on prior information for the vector of parameters in a linear regression model and discuss its properties. Furthermore, we obtain that our new estimator is superior, in the mean square error matrix sense, to the least squares estimator, Liu estimator, ridge estimator and modified ridge estimator. Finally, a numerical example and a Monte Carlo simulation are done to illustrate some of the theoretical results.

Improvement of the Liu Estimator in Weighted Mixed Regression

In this article, we consider the estimation for the vector of parameters in a linear regression model by unifying the sample and the prior information. A new Liu-type biased estimator called weighted mixed Liu estimator is proposed. Furthermore, necessary and sufficient conditions for the superiority of the weighted mixed Liu estimator over the Liu estimator and the weighted mixed estimator proposed by Schaffrin and Toutenburg (1990) in the mean squared error matrix sense are derived. Finally, a numerical example and a Monte Carlo simulation study are given to illustrate some of the theoretical results.

A new stochastic mixed ridge estimator in linear regression model

This paper is concerned with the parameter estimation in linear regression model with additional stochastic linear restrictions. To overcome the multicollinearity problem, a new stochastic mixed ridge estimator is proposed and its efficiency is discussed. Necessary and sufficient conditions for the superiority of the stochastic mixed ridge estimator over the ridge estimator and the mixed estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct and are not correct. Finally, a numerical example is also given to show the theoretical results.