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# Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables

## Abstract

This paper considers the maximum likelihood estimation of a stochastic frontier production function with an interval outcome. We derive an analytical formula for calculating the likelihood function of interval stochastic frontier models. Monte Carlo experiments reveal that the finite sample performance of our method is promising even when the sample size is relatively moderate. We also provide an exact formula for evaluating technical efficiency with interval outcome and apply our method to measure information inefficiency in the labor market for newly graduated college students in Taiwan.

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## Notes

1. 1.

Other distribution assumptions for u have also been discussed in the literature, including truncated normal distribution (Stevenson 1980), gamma distribution (Greene 1990), and Weibull distribution (Tsionas 2007). Recently, Badade and Ramanathan (2020) propose a probabilistic frontier model using a logit model specification.

2. 2.

For discussion and evaluation of c1 and c2, readers are referred to page 261 of Tsay et al. (2013).

3. 3.

The original survey conducted by Peng (2005) has nine categories for worker’s monthly wage: (1) less than 15,840, (2) between 15,840 and 22,800, (3) between 22,800 and 28,800, (4) between 28,800 and 36,300, (5) between 36,300 and 45,800, (6) between 45,800 and 57,800, (7) between 57,800 and 72,800, (8) between 72,800 and 83,900, and (9) more than 83,900. However, some categories contain only small amount of observations. For example, there are only 39 observations in the first category and only 213 observations in total in categories (6)–(9). To keep the number of observations in each category more balanced, we merge the first two categories, and we combine categories (5)–(9) into one.

4. 4.

The efficiency estimates certainly depend on the specification of model used for empirical analysis and the distribution assumption imposed on composite error. Therefore, we treat our empirical results as a starting point of analyzing the market inefficiency of recent Taiwanese college graduates. More empirical works and data collection are useful to enhance our knowledge about this important issue.

## References

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## Acknowledgements

We thank for the valuable comments from Peter Schmidt of Michigan State University and the participants of the DEAIC2017 conference at Hefei, China.

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Correspondence to Shih-Tang Hwu, Tsu-Tan Fu or Wen-Jen Tsay.

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## Appendix

### Appendix

For convenience of exposition, we suppress subindex throughout the Appendix, and two equations from (Abramowitz and Stegun, 1970, Eqs. 7.11 and 7.4.32) are given:

$$Erf(z)=\frac{2}{\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{z}}\exp (-{t}^{2})=2{\displaystyle\int\nolimits_{0}^{\sqrt{2}z}}\phi (t)dt,$$
$$\displaystyle\int \exp -\left(k{x}^{2}+2mx+n\right)dx=\frac{1}{2}\sqrt{\frac{\pi }{k}}\exp \left(\frac{{m}^{2}-kn}{k}\right)Erf\left(\sqrt{k}x+\frac{m}{\sqrt{k}}\right)+C,\ k\,\ne \,0,$$

where C denotes a finite constant.

### Derivation of Proposition 1

Given that (Q, a, b) ∈ R, b > 0, Erf(− x) = − Erf(x), and define $$\varepsilon =\sqrt{2}v$$, we have:

$$\begin{array}{ll}I(Q,a,b)&=\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left({\displaystyle\int\nolimits_{-\infty }^{\sqrt{2}av}}\phi \left(\zeta \right)d\zeta \right)\phi \left(\sqrt{2}bv\right)dv\\ &=\frac{\sqrt{2}}{2}{\displaystyle\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\end{array}$$

Tsay et al. (2013) show that Erf(x) can be well approximated by a function, $$g(x)=1-\exp \left({c}_{1}x+{c}_{2}{x}^{2}\right)$$ for x ≥ 0, where c1 and c2 are discussed in “Interval stochastic frontier model.” We divide the derivation into four cases: (Q ≥ 0, a ≥ 0), (Q ≤ 0, a ≥ 0), (Q ≥ 0, a ≤ 0), and (Q ≤ 0, a ≤ 0).

Case 1. (Q ≥ 0, a ≥ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,=\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\quad\quad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\infty }}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\quad\quad\,\,\approx \frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\infty }}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad-\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\quad\quad\,\,=\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\infty }}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad-\,\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx \frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1-Erf\left(\frac{-a{c}_{1}+\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}Erf\left(\frac{bQ}{\sqrt{2}}\right).\end{array}$$

Case 2. (Q ≤ 0, a ≥ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx \frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp \left(-{b}^{2}{v}^{2}\right)dv\\ \qquad\qquad\,\,=\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{\infty }}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$I(Q,a,b)\approx \frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1-Erf\left(\frac{-a{c}_{1}-\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}.$$

Case 3. (Q ≥ 0, a ≤ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{0}^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\left(1-Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx -\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{0}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\,\,=-\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{0}}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\frac{Q}{\sqrt{2}}}^{0}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx -\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1+Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}\\ \qquad\qquad\quad\,\,-\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)+Erf\left(\frac{-a{c}_{1}-\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}.\end{array}$$

Case 4. (Q ≤ 0, a ≤ 0):

$$\begin{array}{ll}I(Q,a,b)\,=\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\left(1+Erf\left(av\right)\right)\phi \left(\sqrt{2}bv\right)dv\\ \qquad\qquad\,\,\approx -\frac{\sqrt{2}}{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp \left({c}_{1}av+{c}_{2}{a}^{2}{v}^{2}\right)\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\qquad+\,\sqrt{2}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\frac{1}{\sqrt{2\pi }}\exp (-{b}^{2}{v}^{2})dv\\ \qquad\qquad\,\,=-\frac{1}{2\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp \left(-({b}^{2}-{a}^{2}{c}_{2}){v}^{2}+a{c}_{1}v\right)dv\\ \qquad\qquad\qquad+\,\frac{1}{\sqrt{\pi }}{\displaystyle\int\nolimits_{-\infty }^{\frac{Q}{\sqrt{2}}}}\exp (-{b}^{2}{v}^{2})dv.\end{array}$$

When we use (7.4.32) of Abramowitz and Stegun (1970), I(Q, a, b) in this case can be approximated by:

$$\begin{array}{ll}I(Q,a,b)\,\approx -\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[1+Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}\\ \qquad\qquad\quad\,\,-\frac{\exp \left(\frac{{a}^{2}{c}_{1}^{2}}{4{b}^{2}-4{a}^{2}{c}_{2}}\right)\left[Erf\left(\frac{\sqrt{{a}^{2}}{c}_{1}}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)+Erf\left(\frac{-a{c}_{1}+\sqrt{2}Q({b}^{2}-{a}^{2}{c}_{2})}{2\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}\right)\right]}{4\sqrt{{b}^{2}-{a}^{2}{c}_{2}}}+\frac{1}{2b}Erf\left(\frac{bQ}{\sqrt{2}}\right).\end{array}$$

Combining the results in Cases 1–4, we obtain the result in Proposition 1.

### Derivation of Proposition 3

$$\begin{array}{ll}&E\left(u| a\,<\,\varepsilon \,<\, b\right)\\ &=E\left(E\left(u| \varepsilon \right)| a\,<\,\varepsilon \,<\, b\right)\\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}E\left(u| \varepsilon \right)f\left(\varepsilon \right)d\varepsilon \\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\left(S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\varepsilon +\frac{{\sigma }_{u}{\sigma }_{v}}{\sigma }\frac{\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)}{{{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)}\right)\left(\frac{2}{\sigma }\phi \left(\frac{\varepsilon }{\sigma }\right){{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)\right)d\varepsilon \\ &=\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}\left[{\displaystyle\int\nolimits_{a}^{b}}S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\varepsilon \frac{2}{\sigma }\phi \left(\frac{\varepsilon }{\sigma }\right){{\Phi }}\left(S\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon +{\displaystyle\int\nolimits_{a}^{b}}\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\phi \left(\frac{\varepsilon }{\sigma }\right)\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon \right]\\ &={E}_{1}+{E}_{2}.\end{array}$$

We first rewrite E1 as:

$$\begin{array}{ll}{E}_{1}=\,\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}\frac{S{\sigma }_{u}^{2}}{{\sigma }^{2}}{\displaystyle\int\nolimits_{a}^{b}}\varepsilon \ f\left(\varepsilon \right)d\varepsilon \\ \quad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}E\left(\varepsilon | a\,< \,\varepsilon \,<\, b\right).\end{array}$$

As we can see, the integral part in E1 is a censored conditional expectation of $$f\left({\varepsilon }_{}\right)$$ given that ε ∈ (a, b). Using Proposition 1 in Flecher et al. (2009), we can evaluate E1 as:

$${E}_{1}=S\frac{{\sigma }_{u}^{2}}{\sigma }\left\{-\sigma \frac{{\left[f(\varepsilon )\right]}_{a}^{b}}{{\left[F(\varepsilon )\right]}_{a}^{b}}+\frac{2\lambda }{\sqrt{2\pi }{\lambda }^{* }}\frac{{\left[{{\Phi }}\left(\frac{S{\lambda }^{* }}{\sigma }\varepsilon \right)\right]}_{a}^{b}}{{\left[F(\varepsilon )\right]}_{a}^{b}}\right\},$$

where $${\lambda }^{* }=\sqrt{1+{\lambda }^{2}}$$.

We can also recast E2 as:

$$\begin{array}{ll}{E}_{2}=\,\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\phi \left(\frac{\varepsilon }{\sigma }\right)\phi \left(\lambda \frac{\varepsilon }{\sigma }\right)d\varepsilon \\ \quad\,\,=\frac{2{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\frac{1}{2\pi }\exp \left[-\frac{1}{2}{\left(\frac{\varepsilon }{\sigma }\right)}^{2}\right]\exp \left[-\frac{{\lambda }^{2}}{2}{\left(\frac{\varepsilon }{\sigma }\right)}^{2}\right]d\varepsilon \\ \quad\,\,=\frac{{\sigma }_{u}{\sigma }_{v}}{\pi {\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\displaystyle\int\nolimits_{a}^{b}}\exp \left(-\frac{{{\lambda }^{* }}^{2}{\varepsilon }^{2}}{2{\sigma }^{2}}\right)d\varepsilon .\end{array}$$

Using (7.4.32) of Abramowitz and Stegun (1970), we can evaluate E2 as:

$$\begin{array}{ll}{E}_{2}=\frac{{\sigma }_{u}{\sigma }_{v}}{\pi {\sigma }^{2}}\frac{1}{{[F\left(\varepsilon \right)]}_{a}^{b}}{\left[\frac{1}{2}\sqrt{\frac{\pi }{\frac{{{\lambda }^{* }}^{2}}{2{\sigma }^{2}}}}erf\left(\frac{{\lambda }^{* }}{\sqrt{2}\sigma }\varepsilon \right)\right]}_{a}^{b}\\ \quad\,\,=\frac{{\sigma }_{u}{\sigma }_{v}}{\sqrt{2\pi }\sigma {\lambda }^{* }}\frac{{\left[erf\left(\frac{{\lambda }^{* }}{\sqrt{2}\sigma }\varepsilon \right)\right]}_{a}^{b}}{{\left[F\left(\varepsilon \right)\right]}_{a}^{b}}.\end{array}$$

With the above results, we now obtain the JLMS interval efficiency estimate.

### Proof of Corollary 1

We express the limit of Proposition 3 as:

$$\begin{array}{ll}\lim\limits_{\delta \to 0}E\left(u| c\,<\, \varepsilon \,<\,c+\delta \right)\,=\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{\sigma }\left\{-\sigma \frac{{\left[f\left(\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}+\frac{2\lambda }{\sqrt{2\pi }{\lambda }^{* }}\frac{{\left[{{\Phi }}\left(\frac{S{\lambda }^{* }}{\sigma }\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}\right\}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}\frac{\frac{{\sigma }_{u}{\sigma }_{v}}{\sqrt{2\pi }\sigma {\lambda }^{* }}{\left[erf\left(\frac{{\lambda }^{* }}{\sqrt{2}{\sigma }^{2}}\varepsilon \right)\right]}_{c}^{c+\delta }}{{\left[F\left(\varepsilon \right)\right]}_{c}^{c+\delta }}\\ \qquad\qquad\qquad\qquad\qquad\,\,=\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{\sigma }\frac{-\frac{2S\lambda }{\sigma }\phi \left(\frac{c+\delta }{\sigma }\right)\phi \left(\lambda \frac{c+\delta }{\sigma }\right)-\frac{2c}{{\sigma }^{2}}\phi \left(\frac{c+\delta }{\sigma }\right){{\Phi }}\left(\lambda \frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}\frac{2S\lambda }{\sqrt{2\pi }}\frac{\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\lim\limits_{\delta \to 0}\frac{2}{\sqrt{2\pi }}\frac{{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)}{f\left(c+\delta \right)}\\ \qquad\qquad\qquad\qquad\qquad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}c-\frac{\frac{2{S}^{2}\lambda }{\sigma }\phi \left(\frac{c}{\sigma }\right)\phi \left(\lambda \frac{c}{\sigma }\right)-\frac{2{S}^{2}\lambda }{\sqrt{2\pi }\sigma }\phi \left({\lambda }^{* }\frac{c}{\sigma }\right)}{f\left(c\right)}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\frac{2}{\sqrt{2\pi }}\frac{{\sigma }_{u}{\sigma }_{v}}{{\sigma }^{2}}\frac{\phi \left({\lambda }^{* }\frac{c}{\sigma }\right)}{f\left(c\right)}\\ \qquad\qquad\qquad\qquad\qquad\,\,=S\frac{{\sigma }_{u}^{2}}{{\sigma }^{2}}c+\frac{{\sigma }_{u}{\sigma }_{v}}{\sigma }\frac{\phi \left(\lambda \frac{c}{\sigma }\right)}{{{\Phi }}\left(S\lambda \frac{c}{\sigma }\right)},\end{array}$$

where the second equality is based on L’hospital Rule, and the fourth equality follows from the observation that $$\phi \left(\frac{c+\delta }{\sigma }\right)\phi \left(\lambda \frac{c+\delta }{\sigma }\right)=\frac{1}{\sqrt{2\pi }}\phi \left({\lambda }^{* }\frac{c+\delta }{\sigma }\right)$$.

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Hwu, ST., Fu, TT. & Tsay, WJ. Estimation and efficiency evaluation of stochastic frontier models with interval dependent variables. J Prod Anal 56, 33–44 (2021). https://doi.org/10.1007/s11123-021-00609-w

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### Keywords

• Stochastic frontier analysis
• Interval dependent variable
• Technical efficiency

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