On Optimal and Asymptotic Properties of a Fuzzy L₂ Estimator

Round 1

Reviewer 1 Report

The manuscript generalizes the most common estimator in linear regression models $\hat{\beta} = (X^{\prime} X)^{-1} X^{\prime} Y$ for fuzzy input - fuzzy output linear regression models. Only triangular fuzzy numbers are considered in the paper. The given estimator is formally generalized for fuzzy-valued matrices. Some statistical properties are established.   Unfortunately, the authors initially consider only the models $y= \beta_1 x_1 + \ldots + \beta_p x_p + e$ where $\beta_1 \ge 0, \ldots, \beta_p \ge 0$, $x_1 \ge 0, \ldots, x_p \ge 0$ (if the fuzziness is excluded). It makes the paper insignificant. The authors do not consider the problem of subtraction of fuzzy numbers that is the most difficult. The Assumption B is restrictive.   Under high quality of the presentation such results may be published in a strictly specialized journal on fuzzy logic but not in a universally recognized mathematical journal.   Unfortunately, the quality of the presentation is not high in the manuscript. For instance, it is not explained what is it "a family of all triangular fuzzy numbers defined on the non-negative real numbers" (page 2). It is not indicated that $X_{ij} \in \mathcal{F}_{T}$ (page 4).

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Reviewer 2 Report

Interesting work that presents numerous theories, assumptions and evidence. However, in my opinion, the presented theories should be highlighted in the text. In a situation where the authors refer to the theory, e.g. 2.3, the reader must in fact guess which statement is meant. Of course, this is a purely editorial comment. Moreover, despite the theoretical nature of the work, it seems that the presented simulation results should be discussed in more detail.

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Reviewer 3 Report

Notation error on line 49. Confusing notation on line 92. The expression (n x p +1) is confusing. Notation error on line 94, does not match the notation of lines 105, 106 and 107. The format of sections 4 and 5 are different from section 6, and make them difficult to read. The numbering of the theorems, lemmas or corollaries does not appear.

Author Response

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