Abstract
In this paper, we estimate the parameters of fuzzy regression models and investigate a statistical inferences with crisp inputs and fuzzy outputs for each \(\alpha \)-cut. The proposed approaches of statistical inferences are fuzzy least squares (FLS) method and bootstrap technique. FLS is constructed on the basis of minimizing the sum of square of the total difference between observed and estimated outputs. Numerical examples are illustrated to perform the hypotheses test and to provide the percentile confidence regions by proposed approach.
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Acknowledgments
This research was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant number: NRF-2014R1A1A2002032).
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Communicated by J.-W. Jung.
Appendix
Appendix
To prove Eq. (6), the following four cases apply according to endpoints of \({\hat{y}}_{i_{\alpha }}\), \({\hat{y}}_{i_{\alpha }}^{(b)}\).
-
1.
\({\hat{y}}_{i_{\alpha }}^{L} < {\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{(b)L} < {\hat{y}}_{i_{\alpha }}^{(b)U}\).
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2.
\({\hat{y}}_{i_{\alpha }}^{L}>{\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{(b)L} < {\hat{y}}_{i_{\alpha }}^{(b)U}\).
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3.
\({\hat{y}}_{i_{\alpha }}^{L}<{\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{(b)L} > {\hat{y}}_{i_{\alpha }}^{(b)U}\).
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4.
\({\hat{y}}_{i_{\alpha }}^{L}>{\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{(b)L} > {\hat{y}}_{i_{\alpha }}^{(b)U}\).
In Case 1, in the given conditions, it follows that
\({\hat{y}}_{i_{\alpha }}^{-}={\hat{y}}_{i_{\alpha }}^{L}\) and \({\hat{y}}_{i_{\alpha }}^{+}={\hat{y}}_{i_{\alpha }}^{U}\) so that \({\hat{y}}_{i_{\alpha }}^{(b)-}={\hat{y}}_{i_{\alpha }}^{L}+e_{i_{\alpha }}^{(b)-}\) and \({\hat{y}}_{i_{\alpha }}^{(b)+}={\hat{y}}_{i_{\alpha }}^{U}+e_{i_{\alpha }}^{(b)+}\).
In Case 2, in the given conditions, it follows that
\({\hat{y}}_{i_{\alpha }}^{-}={\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{+}={\hat{y}}_{i_{\alpha }}^{L}\) so that \({\hat{y}}_{i_{\alpha }}^{(b)-}={\hat{y}}_{i_{\alpha }}^{U}+e_{i_{\alpha }}^{(b)-}\) and
\({\hat{y}}_{i_{\alpha }}^{(b)+}={\hat{y}}_{i_{\alpha }}^{L}+e_{i_{\alpha }}^{(b)+}\).
In Cases 1, 2, taking \({\hat{y}}_{i_{\alpha }}^{(b)}\ominus _g {\hat{y}}_{i_{\alpha }}\), produces, \({\hat{y}}_{i_{\alpha }}^{(b)-}-{\hat{y}}_{i_{\alpha }}^{-}=e_{i_{\alpha }}^{(b)-}\) and \({\hat{y}}_{i_{\alpha }}^{(b)+}-{\hat{y}}_{i_{\alpha }}^{+}=e_{i_{\alpha }}^{(b)+}\). Therefore
The intercept coefficient is derived in a similar way.
In Case 3. In the given conditions, it follows that
\({\hat{y}}_{i_{\alpha }}^{-}={\hat{y}}_{i_{\alpha }}^{L}\) and \({\hat{y}}_{i_{\alpha }}^{+}={\hat{y}}_{i_{\alpha }}^{U}\) so that \({\hat{y}}_{i_{\alpha }}^{(b)-}={\hat{y}}_{i_{\alpha }}^{U}+e_{i_{\alpha }}^{(b)+}\) and \(\quad {\hat{y}}_{i_{\alpha }}^{(b)+}={\hat{y}}_{i_{\alpha }}^{L}+e_{i_{\alpha }}^{(b)-}\).
In Case 4. In the given conditions, it follows that
\({\hat{y}}_{i_{\alpha }}^{-}={\hat{y}}_{i_{\alpha }}^{U}\) and \({\hat{y}}_{i_{\alpha }}^{+}={\hat{y}}_{i_{\alpha }}^{L}\) so that \({\hat{y}}_{i_{\alpha }}^{(b)-}={\hat{y}}_{i_{\alpha }}^{L}+e_{i_{\alpha }}^{(b)+}\) and \(\quad {\hat{y}}_{i_{\alpha }}^{(b)+}={\hat{y}}_{i_{\alpha }}^{U}+e_{i_{\alpha }}^{(b)-}\).
From the conditions, Case 3 and Case 4 impossible.
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Lee, WJ., Jung, H.Y., Yoon, J.H. et al. The statistical inferences of fuzzy regression based on bootstrap techniques. Soft Comput 19, 883–890 (2015). https://doi.org/10.1007/s00500-014-1415-5
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DOI: https://doi.org/10.1007/s00500-014-1415-5