The ‘wrong skewness’ problem: a re-specification of stochastic frontiers

Abstract

In this paper, we study the ‘wrong skewness phenomenon’ in stochastic frontiers (SF), which consists in the observed difference between the expected and estimated sign of the asymmetry of the composite error, and causes the ‘wrong skewness problem’, for which the estimated inefficiency in the whole industry is zero. We propose a more general and flexible specification of the SF model, introducing dependences between the two error components and asymmetry (positive or negative) of the random error. This re-specification allows us to decompose the third moment of the composite error into three components, namely: (i) the asymmetry of the inefficiency term; (ii) the asymmetry of the random error; and (iii) the structure of dependence between the error components. This decomposition suggests that the wrong skewness anomaly is an ill-posed problem, because we cannot establish ex ante the expected sign of the asymmetry of the composite error. We report a relevant special case that allows us to estimate the three components of the asymmetry of the composite error and, consequently, to interpret the estimated sign. We present two empirical applications. In the first dataset, where the classic SF has the wrong skewness, an estimation of our model rejects the dependence hypothesis, but accepts the asymmetry of the random error, thus justifying the sign of the skewness of the composite error. More importantly, we estimate a non-zero inefficiency, thus solving the wrong skewness problem. In the second dataset, where the classic SF does not yield any anomaly, an estimation of our model provides evidence for the presence of dependence. In such situations, we show that there is a remarkable difference in the efficiency distribution between the classic SF and our class of models.

Appendices

Appendix 1

1.1 Proof of proposition 1

In order to prove Proposition 1 easily, we report some preliminary results in the following Lemma.

Lemma 1

1. U ~ Exp(δ _u ) then

• r-th moment is $E\left( {{U^r}} \right) = \delta _u^r{\it{\Gamma }}\left( {r + 1} \right)$. Consequently, we have: E(U) = δ _u , $E\left( {{U^2}} \right) = 2\delta _u^2$ and $E\left( {{U^3}} \right) = 6\delta _u^3$.

• Denoted with $F(u) = 1 - {e^{ - \frac{u}{{{\delta _u}}}}}$ the distribution function of the random variable U, after algebra, we obtain $E\left[ {{U^r}F\left( U \right)} \right] = E\left( {{U^r}} \right)\left( {1 - \frac{1}{{{2^{r + 1}}}}} \right)$

• 2. If V ~ GL(α _v ,δ _v ), with pdf ${g_V}\left( v \right) = \frac{{{\alpha _v}}}{{{\delta _v}}}{e^{ - \frac{{v + {\delta _v}\left[ {\Psi \left( {{\alpha _v}} \right) - \Psi \left( 1 \right)} \right]}}{{{\delta _v}}}}}{\left( {1 + {e^{ - \frac{{v + {\delta _v}\left[ {\Psi \left( {{\alpha _v}} \right) - \Psi \left( 1 \right)} \right]}}{{{\delta _v}}}}}} \right)^{ - {\alpha _v} - 1}}$ then

• E(V) = 0;

• $E\left( {{V^2}} \right) = Var\left( V \right) = \delta _v^2\left[ {\Psi \prime \left( {{\alpha _v}} \right) + \Psi \prime \left( 1 \right)} \right]$;

• $E\left( {{V^3}} \right) = \delta _v^3\left[ {\Psi \prime\prime \left( {{\alpha _v}} \right) - \Psi \prime\prime \left( 1 \right)} \right]$.

• Denoted with ${G_V}(v) = {\left( {1 + {e^{ - \frac{{v + {\delta _v}\left[ {\Psi \left( {{\alpha _v}} \right) - \Psi \left( 1 \right)} \right]}}{{{\delta _v}}}}}} \right)^{ - {\alpha _v}}}$ the distribution function of the random variable V, it is easy to prove that E[V ^k G(V)] = (1/2)E{[δ _v (Ψ(2α _v )−Ψ(α _v )) + V]^k|2α _v ,δ _v }, where E[.|2α _v ,δ _v ] is the expectation with respect to the GL with parameters 2α _v and δ _v . In particular, for k = 1 and k = 2 we have $E\left[ {VG\left( V \right)} \right] = \frac{{{\delta _v}}}{2}\left[ {\Psi \left( {2{\alpha _v}} \right) - \Psi \left( {{\alpha _v}} \right)} \right]$ and $E\left[ {{V^2}G\left( V \right)} \right] = \frac{{\delta _v^2}}{2}\left\{ {{{\left[ {\Psi \left( {2{\alpha _v}} \right) - \Psi \left( {{\alpha _v}} \right)} \right]}^2} + \left[ {\Psi \prime \left( {2{\alpha _v}} \right) - \Psi \prime \left( 1 \right)} \right]} \right\}\hfill$, respectively.

• 3. if (U,V) ~ f _U,V (u,v) = f _U (u)g _V (v)[1 + θ(1−2F _U (u)) (1−2G _V (v))] then

$$ E\left( {{U^r}{V^k}} \right) = \left( {1 + \theta } \right)E\left( {{U^r}} \right)E\left( {{V^k}} \right) \hfill \\ \\ - 2\theta \left\{ {E\left( {{U^r}} \right)E\left[ {{V^k}G\left( V \right)} \right] + E\left[ {{U^r}F\left( U \right)} \right]E\left( {{V^k}} \right) - } \right. \hfill \\ \\ \left. {2E\left[ {{U^r}F\left( U \right)} \right]E\left[ {{V^k}G\left( V \right)} \right]} \right\} = E\left( {{U^r}} \right)E\left( {{V^k}} \right) + \hfill \\ \\ \theta \left( {\frac{1}{{{2^r}}} - 1} \right)E\left( {{U^r}} \right)\left\{ {E\left( {{V^k}} \right) - E\left({V^k}{G_V}\right.\left( V \right)} \hfill \right\} $$

Now, we can prove the Proposition 1.

• 1. The pdf of composite error is ${f_{\cal E}}\left( \epsilon \right) = {\int}_{{\Re ^ + }} {f_{U,V}}\left( {u,\epsilon + u} \right)du$ where f _U,V (u,ϵ + u) = f _U (u) _V (ϵ + u)c(F _V (u),G _V (ϵ + u)).

  Given that c(⋅, ⋅) is a density copula of a FGM copula, we have

  $$ {f_{U,V}}\left( {u,\epsilon + u} \right) = \left( {1 + \theta } \right){f_U}\left( u \right){g_V}\left( {\epsilon + u} \right) - 2\theta {f_U}(u)\\ \\ {g_V}\left( {\epsilon + u} \right) {G_V}\left( {\epsilon + u} \right)- 2\theta {f_U}(u){g_V}\left( {\epsilon + u} \right){F_U}\left( u \right)\\ \\ + 4\theta {f_U}(u){g_V}\left( {\epsilon + u} \right){F_U}\left( u \right){G_V}\left( {\epsilon + u} \right)\\ $$

  (14)

  Using (14), we have ${f_{\cal E}}\left( \epsilon \right) = \left( {1 + \theta } \right){I_1} - 2\theta \left\{ {{I_2} + {I_3} - 2I} \right\}$, where $I = {\int}_{{\Re ^ + }} {{f_U}\left( u \right){g_V}\left( {\epsilon + u} \right){F_U}\left( u \right){G_V}\left( {\epsilon + u} \right)du}$, and I _i , for i = 1,2,3 are special cases of I.

  Now, in order to calculate the integral I, we observe that

  $${f_U}\left( u \right){g_V}\left( {\epsilon + u} \right){F_U}\left( u \right){G_V}\left( {\epsilon + u} \right)\kern6pc \\ \\ = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _u}{\delta _v}}}{e^{ - \frac{u}{{{\delta _u}}} - \frac{u}{{{\delta _v}}}}}\left( {1 - {e^{ - \frac{u}{{{\delta _u}}}}}} \right){\left( {1 + {k_1}\left( \epsilon \right){e^{ - \frac{u}{{{\delta _u}}}}}} \right)^{ - 2{\alpha _v} - 1}}\\ $$

  (15)

  where ${k_1}\left( \epsilon \right) = {e^{ - \frac{{\epsilon + {\delta _v}\left[ {\Psi \left( {{\alpha _v}} \right) - \Psi \left( 1 \right)} \right]}}{{{\delta _v}}}}}$. After algebra, we can write

  $$I = \frac{{\alpha _v}{k_1}(\epsilon )^{-2{\alpha _v}}}{{\delta _u}{\delta _v}}\left\{ {\int\limits_{\Re^+}} ({e^{- u}})^{\frac{1}{\delta _u} + \frac{1}{{\delta _v}}}[ 1 + {k_1}(\epsilon )({e^{- u}})^{\frac{1}{{{\delta _v}}}} ]^{ - 2{\alpha _v} - 1}du - \right.\\ \left. - \mathop {\int}\limits_{\Re ^ + } ({e^{- u}} )^{\frac{2}{\delta _u} + \frac{1}{\delta _v}}[1 + {k_1}( \epsilon )({e^{- u}})^{\frac{1}{\delta _v}}]^{- 2{\alpha _v} - 1}du \right\}$$

  If before we put y = e ^−u and then $t = {y^{\frac{1}{{{\delta _v}}}}}$, after algebra, we obtain

  $$I = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _u}}}\left\{ {\mathop {\int}\limits_0^1 {{t^{\frac{{{\delta _v}}}{{{\delta _u}}}}}{{\left( {1 + {k_1}\left( \epsilon \right)t} \right)}^{ - 2{\alpha _v} - 1}}dt} - \mathop {\int}\limits_0^1 {{t^{2\frac{{{\delta _v}}}{{{\delta _u}}}}}{{\left( {1 + {k_1}\left( \epsilon \right)t} \right)}^{ - 2{\alpha _v} - 1}}dt} } \right\}$$

  Bearing in mind that for hypergeometric function is true the following

  $$\frac{{{\it{\Gamma }}\left( {c - b} \right){\it{\Gamma }}\left( b \right)}}{{{\it{\Gamma }}\left( c \right)}}{{}_2}{F_1}\left( {a,\,b;\,c;\,s} \right) = \mathop {\int}\limits_0^1 {{t^{b - 1}}{{\left( {1 - t} \right)}^{c - b - 1}}{{\left( {1 - st} \right)}^{ - a}}dt} $$

  We obtain

  $$\begin{array}{ccccc}\\ I = & \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _u}}}\left\{ {\frac{1}{{\frac{{{\delta _v}}}{{{\delta _u}}} + 1}}{\,_2}{F_1}\left( {2{\alpha _v} + 1,\frac{{{\delta _v}}}{{{\delta _u}}} + 1;\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right)} \right.\\ \\ & - \left. {\frac{1}{{2\frac{{{\delta _v}}}{{{\delta _u}}} + 1}}{\,_2}{F_1}\left( {2{\alpha _v} + 1,2\frac{{{\delta _v}}}{{{\delta _u}}} + 1;2\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right)} \right\}\\ \end{array}$$

• 2. By Lemma 1, we can to verify that

• E(ϵ) = −E(U) = −δ _u

• $Var\left( \epsilon \right) = Var\left( U \right) + Var\left( V \right) - 2cov\left( {U,V} \right) = \delta _u^2 + \delta _v^2\left[ {\Psi \prime \left( {{\alpha _v}} \right) + \Psi \prime \left( 1 \right)} \right] - 2cov\left( {U,V} \right)$, where $cov\left( {U,V} \right) = \frac{\theta }{2}E\left( U \right)E\left[ {V{G_V}\left( V \right)} \right] = \frac{\theta }{4}{\delta _u}{\delta _v}\left[ {\Psi \left( {2{\alpha _v}} \right) - \Psi \left( {{\alpha _v}} \right)} \right].$

• Moreover, recalling that for a generic random variable, Z, we have E[Z−E(Z)]³ = E(Z ³)−3E(Z ²)E(Z) + 2[E(Z)]³, after simple algebra, $E{\left[ {U - E\left( U \right)} \right]^3} = 2\delta _u^3$ and $E{\left[ {V - E\left( V \right)} \right]^3} = \delta _v^3\left[ {\Psi \prime\prime \left( {{\alpha _v}} \right) - \Psi \prime\prime \left( 1 \right)} \right]$. Moreover, by Lemma, we have:

$$cov\left( {{U^2},V} \right) = E\left[ {{U^2}V} \right] = \frac{3}{4}\theta \delta _u^2{\delta _v}\left[ {\Psi \left( {2{\alpha _v}} \right) - \Psi \left( {{\alpha _v}} \right)} \right]$$

and

$$cov\left( {U,{V^2}} \right) = - \frac{\theta }{4}{\delta _u}\delta _v^2\left\{ {2\left[ {\Psi \prime \left( {{\alpha _v}} \right) + \Psi \prime \left( 1 \right)} \right] - {{\left[ {\Psi \left( {2{\alpha _v}} \right) \kern5pc - \Psi \left( {{\alpha _v}} \right)} \right]}^2} - \left[ {\Psi \prime \left( {2{\alpha _v}} \right) + \Psi \prime \left( 1 \right)} \right]} \right\}$$

by (2), after algebra, we obtain E[ϵ−E(ϵ)]³ as in Eq. (11).

Appendix 2

1.1 The numerical procedure

The estimation of models like those described in Section 2 requires the ability to compute the density of the composite error. Closed-form expressions for this quantity are available only in some few special cases, such as the noteworthy case addressed by Smith (2008). While in the previous section we provided one more example of a closed-form expression, this section is intended to describe the scheme we use to approximate the likelihood (6) starting from a general joint density f _U,V . Our goal is to provide a numerical tool capable of managing different joint distributions for the couple (U,V), thus widening the set of alternatives one can use when defining an SF model.

Our approach is fairly simple. We approximate the convolution of U with V by means of a numerical quadrature. To be more precise, put ${\cal E} = V - U$. Its density function, ${f_{\cal E}}\left( { \cdot ;{{\Theta }}} \right)$, is obtained by the convolution of U with V:

$${f_{\cal E}}\left( {x;{{\Theta }}} \right) = \mathop {\int}\limits_0^\infty {{f_{U,V}}\left( {u,x + u;{{\Theta }}} \right)du} $$

(16)

An explicit evaluation of the integral in (16) is in general infeasible, which has kept a potential range of possible joint densities almost unexplored. However, an approximation of (16) by Gauss-Laguerre quadrature has proved to be easy and effective, and is presented below.

Let us first rewrite (16) as

$${f_{\cal E}}\left( {x;{{\Theta }}} \right) = \mathop {\int}\limits_0^\infty {{e^{ - u}}{g_x}\left( u \right)du} ,$$

(17)

with g _x (u) = e ^u f _U,V (u,x + u). Fix an integer m, which we will refer to as the order of quadrature. For h = 1…,m, let: (i) t _h be the h-th root of the Laguerre polynomial of order m, L _m (u), and (ii) ω _h be defined by the following system of linear equations^{12 , 13}

$$\mathop {\int}\limits_0^\infty {{s^k}{e^{ - s}}ds} = \mathop {\sum}\limits_{h = 1}^n {{\omega _h}t_h^k} \quad k = 1, \ldots ,2m - 1.$$

(18)

Then, we can write

$${f_{\cal E}}\left( {x;{{\Theta }}} \right) \approx \mathop {\sum}\limits_{h = 1}^m {{\omega _h}{g_x}\left( {{t_h}} \right)} .$$

(19)

Inasmuch as the function g _x ( ⋅ ) is Riemann integrable on the interval [0,∞), standard results in numerical analysis ensure the goodness of the approximation.

We can thus approximate the integral appearing in (16) (and its gradient with respect to Θ) by a finite sum, and insert the approximating density function and its gradient into a quasi Newton-like iteration (however, from experience with the normal—half-normal model with the FGM copula, a few initial iterations with the algorithm of Berndt et al. (1974) is highly recommended). As for the order of quadrature, practice with the normal—half-normal case with the FGM copula shows that m = 12 is sufficient to obtain safe approximations. For values of m around 12, computations of the Laguerre nodes and weights require a fraction of a second, and this is needed only once.

Appendix 3

1.1 Calculation of TE scores

Given the Proposition 1, its Proof in Appendix 1 and Eq. (7) in Section 2, we derive the formula to calculate the Technical Efficiency scores TE _Θ for our models.

We can write

$$T{E_\Theta }\! =\! E\left[ {{e^{ - U}}\left| {\cal E} \right. = {\epsilon ^*}} \right] \!= \frac{1}{{{f_{\cal E}}\left( {{\epsilon ^*};{{\Theta }}} \right)}}\mathop {\int}\limits_{{\Re ^ + }} {{e^{ - u}}{f_{U,V}}\left( {u,\epsilon + u;{{\Theta }}} \right)du} $$

(20)

where f _U,V (u,ϵ + u) is derived in Eq. (14).

After algebra, we obtain:

$$\begin{array}{ccccc}\\ T{E_\Theta } & = E\left[ {{e^{ - U}}\left| {\cal E} \right. = \epsilon } \right] \hfill \\ \\ & = \frac{{{{\bar \omega }_1}\left( \epsilon \right){{\overline H }_1}\left( \epsilon \right) + \theta \left[ {{{\overline \omega }_1}\left( \epsilon \right){{\overline H }_1}\left( \epsilon \right) - 2{{\overline \omega }_2}\left( \epsilon \right){{\overline H }_2}\left( \epsilon \right) - 2{{\overline \omega }_3}\left( \epsilon \right){{\overline H }_3}\left( \epsilon \right) + 4{{\overline \omega }_4}\left( \epsilon \right){{\overline H }_4}\left( \epsilon \right)} \right]}}{{{\omega _1}\left( \epsilon \right){H_1}\left( \epsilon \right) - \theta \left[ {{\omega _1}\left( \epsilon \right){H_1}\left( \epsilon \right) - 2{\omega _2}\left( \epsilon \right){H_2}\left( \epsilon \right) - 2{\omega _3}\left( \epsilon \right){H_3}\left( \epsilon \right) + 4{\omega _4}\left( \epsilon \right){H_4}\left( \epsilon \right)} \right]}}\\ \end{array}$$

(21)

where the H−functions represent hypergeometric functions. In particular, we have:

$${\bar H_1}{ = _2}{F_1}\left( {{\alpha _v} + 1,\frac{{{\delta _v}}}{{{\delta _u}}} + {\delta _v} + 1;\frac{{{\delta _v}}}{{{\delta _u}}} + {\delta _v} + 2; - {k_1}\left( \epsilon \right)} \right)$$

$${H_1}{ = _2}{F_1}\left( {{\alpha _v} + 1,\frac{{{\delta _v}}}{{{\delta _u}}} + 1;\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right)$$

$$\begin{array}{ccccc}\\ {\overline H _2} & \\ \\ & =\displaystyle\frac{1}{{\left[ {{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right)} \right]}}{\,_2}{F_1}\left( {{\alpha _v} + 1,{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right) + 1;{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right) + 2; - {k_1}\left( \epsilon \right)} \right)\\ \\ & - \displaystyle\frac{1}{{\left[ {{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right)} \right]}}{\,_2}{F_1}\left( {{\alpha _v} + 1,{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right) + 1;{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right) + 2; - {k_1}\left( \epsilon \right)} \right)\\ \end{array}$$

$$\begin{array}{ccccc}\\ {H_2}& = { _2}{F_1}\left( {{\alpha _v} + 1,2\frac{{{\delta _v}}}{{{\delta _u}}} + 1;2\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right) \hfill \\ \\ {\overline H _3} & = { _2}{F_1}\left( {2{\alpha _v} + 1,\frac{{{\delta _v}}}{{{\delta _u}}} + {\delta _v} + 1;\frac{{{\delta _v}}}{{{\delta _u}}} + {\delta _v} + 2; - {k_1}\left( \epsilon \right)} \right)\\ \\ {H_3} & = {}_{ 2}{F_1}\left( {2{\alpha _v} + 1,\frac{{{\delta _v}}}{{{\delta _u}}} + 1;\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right) \hfill \\ \end{array}$$

$$\begin{array}{ccccc}\\ {\overline H _4} \\ \\ & = \displaystyle\frac{1}{{\left[ {{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right)} \right]}}{\,_2}{F_1}\left( {2{\alpha _v} + 1,{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right) + 1;{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right) + 2; - {k_1}\left( \epsilon \right)} \right) \\ \\ & - \displaystyle\frac{1}{{\left[ {{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right)} \right]}}{\,_2}{F_1}\left( {2{\alpha _v} + 1,{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right) + 1;{\delta _v}\left( {\frac{2}{{{\delta _u}}} + 1} \right) + 2; - {k_1}\left( \epsilon \right)} \right) \\ \\ & {H_4} = {2}{F_1}\left( {2{\alpha _v} + 1,2\frac{{{\delta _v}}}{{{\delta _u}}} + 1;2\frac{{{\delta _v}}}{{{\delta _u}}} + 2; - {k_1}\left( \epsilon \right)} \right)\hfill \\ \end{array}$$

with ${k_1}\left( \epsilon \right) = {e^{ - \frac{{\epsilon + {\delta _v}\left[ {\Psi \left( {{\alpha _v}} \right) - \Psi \left( 1 \right)} \right]}}{{{\delta _v}}}}}$ and the ω−functions are respectively defined as:

$$\begin{array}{*{20}{l}}\\ {{{\overline \omega }_1}\left( \epsilon \right) = {{\overline \omega }_3}\left( \epsilon \right) = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _u}\left[ {{\delta _v}\left( {\frac{1}{{{\delta _u}}} + 1} \right)} \right]}}} \hfill & {{{\overline \omega }_2}\left( \epsilon \right) = {{\overline \omega }_4}\left( \epsilon \right) = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _u}}}} \hfill \\ \\ {{\omega _1}\left( \epsilon \right) = {\omega _3}\left( \epsilon \right) = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{{\delta _v} + {\delta _v}}}} \hfill & {{\omega _2}\left( \epsilon \right) = {\omega _4}\left( \epsilon \right) = \frac{{{\alpha _v}{k_1}\left( \epsilon \right)}}{{2{\delta _v} + {\delta _v}}}} \hfill \\ \end{array}$$

Appendix 4

1.1 Gaussian and Frank copula functions

	Gaussian	Frank
Parameter	θ∈(−1,1)	θ∈(−∞, +∞)\{0}
Density	$\frac{1}{{\sqrt {1 - {\theta ^2}} }}exp\left( {\frac{{2\theta {{\it{\Phi }}^{ - 1}}\left[ {F\left( u \right)} \right]{{\it{\Phi }}^{ - 1}}\left[ {G\left( v \right)} \right] - {\theta ^2}\left( {{{\it{\Phi }}^{ - 1}}{{\left[ {F\left( u \right)} \right]}^2} + {{\it{\Phi }}^{ - 1}}{{\left[ {G\left( v \right)} \right]}^2}} \right)}}{{2\left( {1 - {\theta ^2}} \right)}}} \right)$	$\frac{{\theta \left( {1 - {e^{ - \theta }}} \right){e^{ - \theta \left( {F\left( u \right) + G\left( v \right)} \right)}}}}{{{{\left[ {\left( {1 - {e^{ - \theta }}} \right) - \left( {1 - {e^{ - \theta F\left( u \right)}}} \right)\left( {1 - {e^{ - \theta G\left( v \right)}}} \right)} \right]}^2}}}$
Distribution	Φ(Φ −1[F(u)],Φ −1[G(v)];θ)	$- {\theta ^{ - 1}}{\rm{ln}}\left[ {1 + \frac{{\left( {{e^{ - \theta F\left( u \right)}} - 1} \right)\left( {{e^{ - \theta G\left( v \right)}} - 1} \right)}}{{\left( {{e^{ - \theta }} - 1} \right)}}} \right]$

Legend: Φ is the Standard Normal distribution and Φ ⁻¹ is the inverse function.