Development of alternative stochastic frontier models for estimating time-space prism vertices

Abstract

This paper develops alternative stochastic frontier models (ASFM) for estimating time-space prism vertices with different distributional assumptions for the inefficiency term that takes a non-negative value. The traditional stochastic frontier model (SFM) assumes that the inefficiency term follows a half-normal or exponential distribution. Under those assumptions, most travelers’ home departure/arrival time will be close to prism vertices, which is not necessarily consistent with actual travel behaviors. To avoid this potential problem, the ASFM adopt alternative distributions for the inefficiency term whose density values can decrease monotonously or vary non-monotonously. Quasi-Monte Carlo simulation method is employed to estimate the ASFM without closed-form likelihood expressions. Simulation experiment results show that SFM needs a substantially greater number of Halton draws for consistent estimators than a typical mixed logit model does. The ASFM are estimated based on the travel data of 1454 Shanghai commuters and 2964 Houston commuters. It is found that models with inefficiency term following a half-normal distribution tend to underestimate the origin vertex of morning prism and overestimate the terminal vertex of evening prism over 50 and 30 min for Shanghai and Houston samples, respectively. The empirical results show the importance of choosing an appropriate distributional assumption for the inefficiency term in the SFM for better understanding the relation between individuals’ departure/arrival time and time-space prism vertices. The SFM based on an appropriate distributional assumption can be applied in activity-based models for big cities to better reflect tighter temporal constraints on metropolitan residents and narrower time-space prisms for outdoor activity arrangement.

Appendix

Derivation of Eq. (5)

Since $g(u) = \frac{2}{{\sigma_{u} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{u^{2} }}{{2\sigma_{u}^{2} }}}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma_{u} > 0$; $h(v) = \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{v^{2} }}{{2\sigma_{v}^{2} }}}}$, $\varepsilon = v - Su$;

$$ f(\varepsilon ) = \int_{0}^{ + \infty } {g(u)h(\varepsilon + Su)} du = \int_{0}^{ + \infty } {\frac{1}{{\pi \sigma_{u} \sigma_{v} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{(\varepsilon + Su)^{2} }}{{2\sigma_{v}^{2} }} - \frac{{u^{2} }}{{2\sigma_{u}^{2} }}}} {\kern 1pt} } du; $$

Let $\sigma^{2} = \sigma_{u}^{2} + \sigma_{v}^{2}$, $\lambda = \sigma_{u} /\sigma_{v}$;

$$ \begin{aligned} f(\varepsilon ) & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\sigma_{u}^{2} \varepsilon^{2} + 2\sigma_{u}^{2} \varepsilon \cdot Su + u^{2} \sigma_{u}^{2} + u^{2} \sigma_{v}^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}}} {\kern 1pt} } du = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{u^{2} (\sigma_{u}^{2} + \sigma_{v}^{2} ) + \sigma_{u}^{2} \varepsilon^{2} + 2\sigma_{u}^{2} \varepsilon \cdot Su}}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}}} {\kern 1pt} } du \\ & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\sigma^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}\left[ {u^{2} + \frac{{\sigma_{u}^{2} \varepsilon^{2} + 2\sigma_{u}^{2} \varepsilon \cdot Su}}{{\sigma^{2} }}} \right]}} {\kern 1pt} } du = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\sigma^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}\left[ {\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} + \frac{{\sigma_{u}^{2} \varepsilon^{2} }}{{\sigma^{2} }} - \frac{{\sigma_{u}^{4} \varepsilon^{2} }}{{\sigma^{4} }}} \right]}} {\kern 1pt} } du \\ & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{\sigma^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }} - \frac{{\sigma_{u}^{2} \varepsilon^{2} }}{{\sigma^{2} \sigma_{v}^{2} }}} \right]}} {\kern 1pt} } du = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{\sigma^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}\left( {1 - \frac{{\sigma_{u}^{2} }}{{\sigma^{2} }}} \right)} \right]}} {\kern 1pt} } du \\ & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{\sigma^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}\left( {\frac{{\sigma_{u}^{2} + \sigma_{v}^{2} - \sigma_{u}^{2} }}{{\sigma^{2} }}} \right)} \right]}} {\kern 1pt} } du = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{\sigma^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} + \frac{{\varepsilon^{2} }}{{\sigma^{2} }}} \right]}} {\kern 1pt} } du \\ & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}}} \int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\sigma^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right)^{2} }} {\kern 1pt} } du \\ \end{aligned} $$

Define $\frac{\sigma }{{\sigma_{u} \sigma_{v} }}\left( {u + \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }}} \right) = t$, then $\frac{\sigma }{{\sigma_{u} \sigma_{v} }}du = dt$; $du = \frac{{\sigma_{u} \sigma_{v} }}{\sigma }dt$.

When $u = 0$, $t = \frac{\sigma }{{\sigma_{u} \sigma_{v} }} \cdot \frac{{S\sigma_{u}^{2} \varepsilon }}{{\sigma^{2} }} = \frac{S\varepsilon \lambda }{\sigma }$; when $u \to + \infty$, $t \to + \infty$. Then,

$$ \begin{aligned} f(\varepsilon ) & = \frac{1}{{\pi \sigma_{u} \sigma_{v} }}e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}}} \cdot \int_{{\frac{S\varepsilon \lambda }{\sigma }}}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} \cdot \frac{{\sigma_{u} \sigma_{v} }}{\sigma }{\kern 1pt} } dt = \frac{1}{\pi \sigma }e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}}} \cdot \int_{{\frac{S\varepsilon \lambda }{\sigma }}}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} {\kern 1pt} } dt = \frac{2}{2\pi \sigma }e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}}} \cdot \int_{{\frac{S\varepsilon \lambda }{\sigma }}}^{ + \infty } {e^{{ - \frac{{t^{2} }}{2}}} {\kern 1pt} } dt \\ & = \frac{2}{\sigma } \cdot \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma^{2} }}}} \cdot \int_{{\frac{S\varepsilon \lambda }{\sigma }}}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{{t^{2} }}{2}}} {\kern 1pt} } dt = \frac{2}{\sigma } \cdot \varphi \left( {\frac{\varepsilon }{\sigma }} \right) \cdot \left[ {1 - \varPhi \left( {\frac{S\varepsilon \lambda }{\sigma }} \right)} \right] = \frac{2}{\sigma } \cdot \varphi \left( {\frac{\varepsilon }{\sigma }} \right) \cdot \varPhi \left( { - \frac{S\varepsilon \lambda }{\sigma }} \right) \\ \end{aligned} $$

Derivation of Eq. (6)

Since $g(u) = \frac{1}{{\sigma_{u} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{u}{{\sigma_{u} }}}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma_{u} > 0$, $h(v) = \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{v^{2} }}{{2\sigma_{v}^{2} }}}}$, $\varepsilon = v - Su$;

$$ \begin{aligned} f(\varepsilon ) & = \int_{0}^{ + \infty } {g(u)h(\varepsilon + Su)} du = \int_{0}^{ + \infty } {\frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{(\varepsilon + Su)^{2} }}{{2\sigma_{v}^{2} }} - \frac{u}{{\sigma_{u} }}}} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\varepsilon^{2} + 2\varepsilon Su + u^{2} }}{{2\sigma_{v}^{2} }} - \frac{u}{{\sigma_{u} }}}} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} - \frac{\varepsilon Su}{{\sigma_{v}^{2} }} - \frac{{u^{2} }}{{2\sigma_{v}^{2} }} - \frac{u}{{\sigma_{u} }}}} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{u^{2} }}{{\sigma_{v}^{2} }} + 2\left( {\frac{u}{{\sigma_{u} }} + \frac{\varepsilon Su}{{\sigma_{v}^{2} }}} \right) + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}} \right]}} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{{u^{2} }}{{\sigma_{v}^{2} }} + \frac{2u}{{\sigma_{v} }}\left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right) + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}} \right]}} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left\{ {\left[ {\frac{u}{{\sigma_{v} }} + \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)} \right]^{2} - \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)^{2} + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}} \right\}}} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left\{ {\left[ {\frac{u}{{\sigma_{v} }} + \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)} \right]^{2} - \frac{{\sigma_{v}^{2} }}{{\sigma_{u}^{2} }} - \frac{2\varepsilon S}{{\sigma_{u} }} - \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }} + \frac{{\varepsilon^{2} }}{{\sigma_{v}^{2} }}} \right\}}} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}\int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left\{ {\left[ {\frac{u}{{\sigma_{v} }} + \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)} \right]^{2} - \frac{{\sigma_{v}^{2} }}{{\sigma_{u}^{2} }} - \frac{2\varepsilon S}{{\sigma_{u} }}} \right\}}} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}e^{{\frac{{\sigma_{v}^{2} }}{{2\sigma_{u}^{2} }} + \frac{\varepsilon S}{{\sigma_{u} }}}} \int_{0}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{1}{2}\left[ {\frac{u}{{\sigma_{v} }} + \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)} \right]^{2} }} {\kern 1pt} } du \\ \end{aligned} $$

Let $\frac{u}{{\sigma_{v} }} + \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right) = t$, then $\frac{1}{{\sigma_{v} }}du = dt$;$du = \sigma_{v} dt$.

When $u = 0$, $t = \frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}$; when $u \to + \infty$, $t \to + \infty$. Then,

$$ \begin{aligned} f(\varepsilon ) & = \frac{1}{{\sqrt {2\pi } \sigma_{u} \sigma_{v} }}e^{{\frac{{\sigma_{v}^{2} }}{{2\sigma_{u}^{2} }} + \frac{\varepsilon S}{{\sigma_{u} }}}} \int_{{\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}}}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} {\kern 1pt} } \cdot \sigma_{v} dt = \frac{1}{{\sigma_{u} }} \cdot e^{{\frac{{\sigma_{v}^{2} }}{{2\sigma_{u}^{2} }} + \frac{\varepsilon S}{{\sigma_{u} }}}} \cdot \frac{1}{{\sqrt {2\pi } }}\int_{{\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}}}^{ + \infty } {{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} {\kern 1pt} } dt \\ & = \frac{1}{{\sigma_{u} }} \cdot e^{{\frac{{\sigma_{v}^{2} }}{{2\sigma_{u}^{2} }} + \frac{\varepsilon S}{{\sigma_{u} }}}} \cdot \left( {1 - \varPhi \left( {\frac{{\sigma_{v} }}{{\sigma_{u} }} + \frac{\varepsilon S}{{\sigma_{v} }}} \right)} \right) = \frac{1}{{\sigma_{u} }} \cdot e^{{\frac{{\sigma_{v}^{2} }}{{2\sigma_{u}^{2} }} + \frac{\varepsilon S}{{\sigma_{u} }}}} \cdot \varPhi \left( { - \frac{{\sigma_{v} }}{{\sigma_{u} }} - \frac{\varepsilon S}{{\sigma_{v} }}} \right) \\ \end{aligned} $$

Derivation of Eq. (7)

Since $g(u) = \frac{u}{{\sigma_{u}^{2} }} \cdot \exp \left( { - \frac{{u^{2} }}{{2\sigma_{u}^{2} }}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma_{u} > 0$, $h(v) = \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{v^{2} }}{{2\sigma_{v}^{2} }}}}$, $\varepsilon = v - Su$;

$$ \begin{aligned} f(\varepsilon ) & = \int_{0}^{ + \infty } {g(u)h(\varepsilon + Su)} du = \int_{0}^{ + \infty } {\frac{u}{{\sigma_{u}^{2} }}e^{{ - \frac{{u^{2} }}{{2\sigma_{u}^{2} }}}} \cdot \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{(\varepsilon + Su)^{2} }}{{2\sigma_{v}^{2} }}}} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }}\int_{0}^{ + \infty } {ue^{{ - \frac{{u^{2} }}{{2\sigma_{u}^{2} }} - \frac{{(\varepsilon + Su)^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }}\int_{0}^{ + \infty } {ue^{{ - \frac{1}{2}\left( {\frac{{u^{2} \sigma_{v}^{2} + \sigma_{u}^{2} (\varepsilon + Su)^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}} \right)}} {\kern 1pt} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }}\int_{0}^{ + \infty } {ue^{{ - \frac{1}{2}\left( {\frac{{u^{2} \sigma_{v}^{2} + \sigma_{u}^{2} \varepsilon^{2} + 2\varepsilon Su\sigma_{u}^{2} + u^{2} \sigma_{u}^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}} \right)}} {\kern 1pt} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }}\int_{0}^{ + \infty } {ue^{{ - \frac{1}{2}\left( {\frac{{u^{2} (\sigma_{v}^{2} + \sigma_{u}^{2} ) + 2\varepsilon Su\sigma_{u}^{2} }}{{\sigma_{u}^{2} \sigma_{v}^{2} }}} \right)}} \cdot {\kern 1pt} {\kern 1pt} } e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} \int_{0}^{ + \infty } {ue^{{ - \frac{{\sigma_{v}^{2} + \sigma_{u}^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}\left( {u^{2} + \frac{{2\varepsilon Su\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)}} {\kern 1pt} } du = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} \int_{0}^{ + \infty } {ue^{{ - \frac{{\sigma_{v}^{2} + \sigma_{u}^{2} }}{{2\sigma_{u}^{2} \sigma_{v}^{2} }}\left[ {\left( {u + \frac{{\varepsilon S\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)^{2} - \left( {\frac{{\varepsilon S\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)^{2} } \right]}} {\kern 1pt} } du \\ & = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} \int_{0}^{ + \infty } {ue^{{\frac{{\left( {\frac{{\varepsilon S\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)^{2} }}{{\frac{{2\sigma_{u}^{2} \sigma_{v}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}}}}} \cdot {\kern 1pt} e^{{ - \frac{{\left( {u + \frac{{\varepsilon S\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)^{2} }}{{\frac{{2\sigma_{u}^{2} \sigma_{v}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}}}}} } du \\ \end{aligned} $$

Let $\mu = \frac{{ - \varepsilon S\sigma_{u}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}$, $\sigma_{r}^{2} = \frac{{\sigma_{u}^{2} \sigma_{v}^{2} }}{{\sigma_{u}^{2} + \sigma_{v}^{2} }}$; then,

$$ f(\varepsilon ) = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} \int_{0}^{ + \infty } {ue^{{\frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot {\kern 1pt} e^{{ - \frac{{\left( {u - \mu } \right)^{2} }}{{2\sigma_{r}^{2} }}}} } du. $$

Let $t = \frac{u - \mu }{{\sigma_{r} }}$, then $u = t\sigma_{r} + \mu$; $dt = \frac{{1{\kern 1pt} {\kern 1pt} }}{{\sigma_{r} }}du$.

$$ \begin{aligned} f(\varepsilon ) & = \frac{1}{{\sqrt {2\pi } \sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }}}} {\kern 1pt} \cdot e^{{\frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\left( {t\sigma_{r} + \mu } \right) \cdot {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} } dt \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} {\kern 1pt} \cdot \sigma_{r} \left( {\int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}t\sigma_{r} \cdot {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} } dt + \int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}\mu \cdot {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} } dt} \right) \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}t\sigma_{r} \cdot {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} } dt + \mu {\kern 1pt} \cdot \int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{{t^{2} }}{2}}} } dt} \right) \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\int_{{ - \frac{\mu }{{\sigma_{r} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}t\sigma_{r} \cdot {\kern 1pt} e^{{ - \frac{{t^{2} }}{2}}} } dt + \mu {\kern 1pt} \cdot \varPhi (\frac{\mu }{{\sigma_{r} }})} \right) \\ \end{aligned} $$

Let $u = \frac{{t^{2} }}{2}$, $du = t{\kern 1pt} {\kern 1pt} dt$.

When $t = - \frac{\mu }{{\sigma_{r} }}$, $u = \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}$; when $u \to + \infty$, $t \to + \infty$. Then,

$$ \begin{aligned} f(\varepsilon ) & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\sigma_{r} \int_{{\frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}}^{ + \infty } {\frac{1}{{\sqrt {2\pi } }}{\kern 1pt} e^{ - u} } du + \mu {\kern 1pt} \cdot \varPhi \left( {\frac{\mu }{{\sigma_{r} }}} \right)} \right) \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\sigma_{r} \cdot \frac{1}{{\sqrt {2\pi } }}\left( { - e^{ - u} |_{{\frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}}^{ + \infty } } \right) + \mu {\kern 1pt} \cdot \varPhi \left( {\frac{\mu }{{\sigma_{r} }}} \right)} \right) \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\sigma_{r} \cdot \frac{1}{{\sqrt {2\pi } }}e^{{ - \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} + \mu {\kern 1pt} \cdot \varPhi \left( {\frac{\mu }{{\sigma_{r} }}} \right)} \right) \\ & = \frac{1}{{\sigma_{v} \sigma_{u}^{2} }} \cdot e^{{ - \frac{{\varepsilon^{2} }}{{2\sigma_{v}^{2} }} + \frac{{\mu^{2} }}{{2\sigma_{r}^{2} }}}} \cdot \sigma_{r} \left( {\sigma_{r} \cdot \varphi \left( {\frac{\mu }{{\sigma_{r} }}} \right) + \mu {\kern 1pt} \cdot \varPhi \left( {\frac{\mu }{{\sigma_{r} }}} \right)} \right) \\ \end{aligned} $$

Derivation of Eq. (8)

Since $g(u) = \frac{1}{{u\sigma_{u} }} \cdot \varphi \left( { - \frac{{\ln (u) - u_{l} }}{{\sigma_{u} }}} \right),\sigma_{u} > 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} - \infty < u_{l} < \infty$, $h(v) = \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{v^{2} }}{{2\sigma_{v}^{2} }}}}$, $\varepsilon = v - Su$;

$$ \begin{aligned} f(\varepsilon ) & = \int_{0}^{ + \infty } {g(u)h(\varepsilon + Su)} du \\ & = \int_{0}^{ + \infty } {\frac{1}{{u\sigma_{u} }} \cdot \varphi \left( { - \frac{{\ln (u) - u_{l} }}{{\sigma_{u} }}} \right) \cdot \frac{1}{{\sigma_{v} \sqrt {2\pi } }}{\kern 1pt} {\kern 1pt} {\kern 1pt} e^{{ - \frac{{(\varepsilon + Su)^{2} }}{{2\sigma_{v}^{2} }}}} } du \\ & = \int_{0}^{ + \infty } {\frac{1}{{u\sigma_{u} }} \cdot \varphi \left( { - \frac{{\ln (u) - u_{l} }}{{\sigma_{u} }}} \right) \cdot \frac{1}{{\sigma_{v} }}{\kern 1pt} {\kern 1pt} \varphi \left( {\frac{\varepsilon + Su}{{\sigma_{v} }}} \right){\kern 1pt} } du \\ \end{aligned} $$