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# On Optimal and Asymptotic Properties of a Fuzzy L2 Estimator

by 1,* and
1
Department of Mathematics and Statistics, Sejong University, Seoul 05006, Korea
2
Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Wasaw, Poland
3
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(11), 1956; https://doi.org/10.3390/math8111956
Received: 31 August 2020 / Revised: 14 October 2020 / Accepted: 24 October 2020 / Published: 4 November 2020
A fuzzy least squares estimator in the multiple with fuzzy-input–fuzzy-output linear regression model is considered. The paper provides a formula for the $L2$ estimator of the fuzzy regression model. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. Using the proposed operations, the formula for the variance, provided and this paper, proves that the estimators have several important optimal properties and asymptotic properties: they are Best Linear Unbiased Estimator (BLUE), asymptotic normality and strong consistency. The confidence regions of the coefficient parameters and the asymptotic relative efficiency (ARE) are also discussed. In addition, several examples are provided including a Monte Carlo simulation study showing the validity of the proposed theorems. View Full-Text
MDPI and ACS Style

Yoon, J.H.; Grzegorzewski, P. On Optimal and Asymptotic Properties of a Fuzzy L2 Estimator. Mathematics 2020, 8, 1956. https://doi.org/10.3390/math8111956

AMA Style

Yoon JH, Grzegorzewski P. On Optimal and Asymptotic Properties of a Fuzzy L2 Estimator. Mathematics. 2020; 8(11):1956. https://doi.org/10.3390/math8111956

Chicago/Turabian Style

Yoon, Jin H., and Przemyslaw Grzegorzewski. 2020. "On Optimal and Asymptotic Properties of a Fuzzy L2 Estimator" Mathematics 8, no. 11: 1956. https://doi.org/10.3390/math8111956

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