Abstract
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.
Highlights
IntroductionRegression analysis is commonly perceived as one of the most useful tools in statistical modeling
Regression analysis is commonly perceived as one of the most useful tools in statistical modeling.If the data could be observed precisely, the classical regression appears usually as a sufficient solution.we encounter a lot of situations where the observations cannot be obtained precisely
Applying fuzzy sets proposed by Zadeh [1], Tanaka et al [2] introduced a fuzzy regression analysis
Summary
Regression analysis is commonly perceived as one of the most useful tools in statistical modeling. Mathematics 2020, 8, 1956 any analytic formulas for the desired estimators but they determine the estimates from the normal equations directly (see [3,19]) To overcome these problems the Yoon et al [33,34,35] redefined the mathematical model of a fuzzy linear regression using the so-called triangular fuzzy matrix and suitable operations defined both on triangular fuzzy numbers like on triangular fuzzy matrices. The importance of triangular and trapezoidal fuzzy numbers has been emphasized in [37] This approach enables us to determine fuzzy least squares estimators of the regression parameters in a concise form which is useful for exploring the statistical properties of the estimators. The asymptotic theory for fuzzy multiple regression model is hardly discussed in the paper so far In this contribution we continue the examination of the fuzzy least squares estimator obtained there, focusing on its fundamental finite-sample and asymptotic properties.
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