The statistical inferences of fuzzy regression based on bootstrap techniques

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.

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 ${\hat{y}}_{i_{\alpha }}^{(b)L} < {\hat{y}}_{i_{\alpha }}^{(b)U}$.

• 2. ${\hat{y}}_{i_{\alpha }}^{L}>{\hat{y}}_{i_{\alpha }}^{U}$ and ${\hat{y}}_{i_{\alpha }}^{(b)L} < {\hat{y}}_{i_{\alpha }}^{(b)U}$.

• 3. ${\hat{y}}_{i_{\alpha }}^{L}<{\hat{y}}_{i_{\alpha }}^{U}$ and ${\hat{y}}_{i_{\alpha }}^{(b)L} > {\hat{y}}_{i_{\alpha }}^{(b)U}$.

• 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

$$\begin{aligned}&E\big [ \big \{ {\hat{\beta }}_{1_{\alpha }}^{(b)} \ominus _g {\hat{\beta }}_{1_{\alpha }} \big \} \big ]\\&\quad =E \Bigg [\frac{\sum _{i=1}^{n}(x_i -{\bar{x}})e_{i_{\alpha }}^{(b)-} }{\sum _{i=1}^{n}(x_i -{\bar{x}})^2},\frac{\sum _{i=1}^{n}(x_i -{\bar{x}}) e_{i_{\alpha }}^{(b)+}}{\sum _{i=1}^{n}(x_i -{\bar{x}})^2} \Bigg ]\\&\quad =\left\{ \begin{array}{ll} \frac{\sum _{i=1}^{n}(x_i -{\bar{x}})}{\sum _{i=1}^{n}(x_i -{\bar{x}})^2} E \Big [e_{i_{\alpha }}^{(b)-},\quad e_{i_{\alpha }}^{(b)+}\Big ] &{}\quad \text {if}\ x_i \ge {\bar{x}}, \\ \frac{\sum _{i=1}^{n}(x_i -{\bar{x}})}{\sum _{i=1}^{n}(x_i -{\bar{x}})^2}\Big \{ \{0\} \ominus _g E \Big [-e_{i_{\alpha }}^{(b)+},-e_{i_{\alpha }}^{(b)-}\Big ] \Big \}&{}\quad \text {if}\ x_i <{\bar{x}} \end{array}\right. \\&\quad =\{0\}. \end{aligned}$$

The intercept coefficient is derived in a similar way.

$$\begin{aligned}&E\big [ \big \{ {\hat{\beta }}_{0_{\alpha }}^{(b)} \ominus _g {\hat{\beta }}_{0_{\alpha }} \big \} \big ] \\&\quad =E \bigg [ {\bar{e}}_{i_{\alpha }}^{(b)-}- \big ({\hat{\beta }}_{1_{\alpha }}^{(b)-} - {\hat{\beta }}_{1_{\alpha }}^- \big ){\bar{x}},\quad {\bar{e}}_{i_{\alpha }}^{(b)+} - \big ({\hat{\beta }}_{1_{\alpha }}^{(b)+} - {\hat{\beta }}_{1_{\alpha }}^+ \big ){\bar{x}}\bigg ]\\&\quad =E\bigg [\Big [{\bar{e}}_{i_{\alpha }}^{(b)-}, {\bar{e}}_{i_{\alpha }}^{(b)+} \Big ]+\bigg \{ \big \{ 0\big \} \ominus _g \Big [- {\hat{\beta }}_{1_{\alpha }}^{(b)-}+{\hat{\beta }}_{1_{\alpha }}^{-}, - {\hat{\beta }}_{1_{\alpha }}^{(b)+}+{\hat{\beta }}_{1_{\alpha }}^{+} \Big ]{\bar{x}} \bigg \}\bigg ] \\&\quad =E\Big [{\bar{e}}_{i_{\alpha }}^{(b)}\Big ] \ominus _g E\Big [ \big \{ {\hat{\beta }}_{0_{\alpha }}^{(b)} \ominus _g {\hat{\beta }}_{0_{\alpha }} \big \} \Big ]{\bar{x}}=\big \{0\big \} \end{aligned}$$

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.