Linear estimation in regression analysis using fuzzy prior information

Arnold, Bernhard F.; Stahlecker, Peter

doi:10.1515/rose.1997.5.2.105

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…Задачi оцiнювання невiдомих параметрiв за результатами спостережень дослiджувались в роботах багатьох авторiв [1][2][3][4][5][6][7][8][9]. Задачам оцiнювання параметрiв в умовах невизначеностi присвячено порiвняно мало робiт.…”

Section: вступunclassified

Guaranteed Root Mean Square Estimates of Observations With Unknown Matrices

Nakonechnyi¹,

Kudin²,

Zinko³

et al. 2022

JNAM

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Section: вступunclassified

Guaranteed Root Mean Square Estimates of Observations With Unknown Matrices

Nakonechnyi¹,

Kudin²,

Zinko³

et al. 2022

JNAM

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“…denotes the maximal eigenvalue of the corresponding matrix. The value of the first supremum in (5) is well-known, and the second supremum may be calculated by using the Lagrangian method (see Arnold/Stahlecker, 1997).…”

Section: A = { S E E K + N L ( S -~O ) '~( S -~O )mentioning

confidence: 99%

“…By (6) and the definition of D and T we get Assuming t > 0, setting V = t T;' + qVo + sW-' and minimizing (7) with respect to C E IRkxn we obtain by elementary calculus and by a convexity argument (see also Arnold/Stahlecker, 1997) that is a minimizer of (7) and thus, the linear affine estimator b* = C * y -( C * X -I)Po -C * Y~ is an FBLAE. Note that expression (8) is similar to that of the classical minimax estimator in case of rk(A) = 1 (see e.g., Stahlecker, 1987).…”

Section: A = { S E E K + N L ( S -~O ) '~( S -~O )mentioning

confidence: 99%

“…Assuming furthermore that the covariance matrix V is known and that the prior knowledge on P is given by a nonempty compact subset B of Ftk we arrive at the classical minimax approach (see e.g., Kuks/Olman, 1972;Bunke, 1975a, b;Lauter, 1975;Hoffmann, 1979;Pilz, 1986;Trenklerl Stahlecker, 1987;StahleckerITrenkler, 1991 andRao/Toutenburg, 1995), where the maximum of the risk on B is minimized with respect to C and c. But of course, only vague prior knowledge on ,8 and V may be available and moreover, the model may be misspecified. In Arnold/Stahlecker (1997) an approach is presented how to implement vague prior information about /? and V , where the minimax principle is generalized by means of fuzzy sets originally introduced by Zadeh (1965).…”

Section: Introductionmentioning

confidence: 99%

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Linear Affine Estimation in Misspecified Linear Regression Models Using Fuzzy Prior Information

Arnold

Stahlecker

1998

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Within the framework of a possibly misspecified linear regression model, a generalized minimax approach is presented in order to determine an optimal linear affine estimator of the regression coefficient O. We make use of some fuzzy prior knowledge about P, about the misspecification and about the covariance structure. Here, the objective functions are based on the weighted mean squared error criterion. In case of symmetric prior information sets the optimization problem is, in essence, reduced to the linear term of the estimator. An application to ellipsoidal fuzzy constraints is given, and it is shown that an affine version of the generalized least squares estimator is optimal with respect to a large class of objective functions when there is no information available about P.

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“…Решение задачи линейной оценки наблюдений в рамках векторной модели, когда имеется наблюдаемый выходной сигнал, который определяется линейным оператором входного сигнала с матричными составляющими и аддитивным вектором случайных возмущений, широко используется в современных приложениях. Предметом исследования научных публикаций [1][2][3][4][5][6][7][8][9] стали задачи линейного оценивания при различных схемах наблюдений. В рамках теории линейной регрессии изучались линейные по наблюдениям оценки, в частности несмещенные, что приводило к уравнениям несмещенности, среди решений которых выделяли минимальные по норме, что позволяло минимизировать среднеквадратическую погрешность при наблюдениях с различными статистическими предположениями.…”

Section: Introductionunclassified

Минимаксные Среднеквадратические Оценки Матричных Параметров В Задачах Линейной Регрессии В Условиях Неопределенности

Nakonechnyi¹,

Kudin²,

Zinko³

et al. 2021

PC&I

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The issues of parameter estimation in linear regression problems with random matrix coefficients were researched. Given that random linear functions are observed from unknown matrices with random errors that have unknown correlation matrices, the problems of guaranteed mean square estimation of linear functions of matrices were investigated. The estimates of the upper and lower guaranteed standard errors of linear estimates of observations of linear functions of matrices were obtained in the case when the sets are found, for which the unknown matrices and correlation matrices of observation errors are known. It was proved that for some partial cases such estimates are accurate. Assuming that the sets are bounded, convex and closed, more accurate two-sided estimates have been gained for guaranteed errors. The conditions when the guaranteed mean squared errors approach zero as the number of observations increases were found. The necessary and sufficient conditions for the unbiasedness of linear estimates of linear functions of matrices were provided. The notion of quasi-optimal estimates for linear functions of matrices was introduced, and it was proved that in the class of unbiased estimates, quasi-optimal estimates exist and are unique. For such estimates, the conditions of convergence to zero of the guaranteed mean-square errors were obtained. Also, for linear estimates of unknown matrices, the concept of quasi-minimax estimates was introduced and it was confirmed that they are unbiased. For special sets, which include an unknown matrix and correlation matrices of observation errors, such estimates were expressed through the solution of linear operator equations in a finite-dimensional space. For quasi-minimax estimates under certain assumptions, the form of the guaranteed mean squared error of the unknown matrix was found. It was shown that such errors are limited by the sum of traces of the known matrices. An example of finding a minimax unbiased linear estimation was given for a special type of random matrices that are included in the observation equation.

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Linear estimation in regression analysis using fuzzy prior information

Cited by 6 publications

References 10 publications

Guaranteed Root Mean Square Estimates of Observations With Unknown Matrices

Guaranteed Root Mean Square Estimates of Observations With Unknown Matrices

Linear Affine Estimation in Misspecified Linear Regression Models Using Fuzzy Prior Information

Минимаксные Среднеквадратические Оценки Матричных Параметров В Задачах Линейной Регрессии В Условиях Неопределенности

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