An expectation conditional maximization algorithm for the skew-normal based stochastic frontier model

Abstract

In this paper, a feasible expectation-conditional-maximization (ECM) algorithm is developed for finding the maximum likelihood estimates of parameters of the skew-normal based stochastic frontier model. The closed-form formulas for updating parameters in CM-steps are derived. The proposed methodology is illustrated with simulations and a real data example, where we find the new ECM algorithm outperforms the numerical approach adopted in the previous study.

Appendix A: Proof of Theorem 1

Proof

The proof of (i) is straightforward by using moment generating functions of $Z, \vert Z_0\vert$ and U. For (ii), note that the joint conditional PDF of $\vert Z_0\vert$ and U given $Y=y$ is

$$\begin{aligned} f(z_0, u\mid y)&=\frac{1}{f_{Y}(y)}f(y\mid u, z_0)f_{\vert Z_0 \vert }(z_0) f_U(u)\\&=\frac{1}{f_{Y}(y)}\frac{1}{\vert b \vert \sigma _z\sqrt{2\pi }}\exp \left[ -\frac{(y-\mu -az_0-cu)^2}{2b^2\sigma _z^2}\right] \,\frac{\sqrt{2}}{\sigma _0\sqrt{\pi }}\exp \left[ -\frac{z_0^2}{2\sigma _0^2}\right] \, \frac{\sqrt{2}}{\sigma _u\sqrt{\pi }}\exp \left[ -\frac{u^2}{2\sigma _u^2}\right] \\ &=\frac{1}{f_Y(y)}\frac{ { \sqrt{2}}}{\vert b \vert \sigma _z\sigma_0\sigma _u\pi ^{3/2}}\exp \left\{ -\frac{1}{2} { \left[ \frac{(y-\mu -az_0-cu)^2}{b^2\sigma _z^2}+\frac{z_0^2}{\sigma _0^2}+\frac{u^2}{\sigma _u^2}\right] } \right\} , \end{aligned}$$

where $f(y\mid u, z_0)$ is the conditional PDF of Y given $U=u$, and $\vert Z_0 \vert =z_0$, $f_{\vert Z_0 \vert }(z_0)$ and $f_U(u)$ are marginal PDFs of $\vert Z_0 \vert$ and $\vert U\vert$, respectively. To write

$$\begin{aligned} \frac{(y-\mu -az_0-cu)^2}{b^2\sigma _z^2}+\frac{z_0^2}{\sigma _0^2}+\frac{u^2}{\sigma _u^2} \end{aligned}$$

as a quadratic form of $(z_0, u)^\top$, let $\varvec{\tau }=(\tau _1, \tau _2)^\top$ and $\Sigma ^{-1}=\left( \begin{matrix} d_{11}, &{} d_{12}\\ d_{12}, &{} d_{22}\end{matrix} \right) .$ By comparing coefficients of

$$\begin{aligned} \frac{(y-\mu -az_0-cu)^2}{b^2\sigma _z^2}+\frac{z_0^2}{\sigma _0^2}+\frac{u^2}{\sigma _u^2} \quad\text { and }\quad \left( z_0-\tau _1, u-\tau _2\right) \Sigma ^{-1} \left( \begin{matrix} z_0-\tau _1,\\ u-\tau _2 \end{matrix} \right) , \end{aligned}$$

we have

$$\begin{aligned} \tau _1&= \dfrac{a\sigma _0^2(y-\mu )}{a^2\sigma _0^2+b^2\sigma _z^2+c^2\sigma _u^2}, \quad \tau _2= \dfrac{c\sigma _u^2(y-\mu )}{a^2\sigma _0^2+b^2\sigma _z^2+c^2\sigma _u^2}, \\ d_{11}&=\dfrac{a^2\sigma _0^2+b^2\sigma _z^2}{b^2\sigma _z^2\sigma _0^2}, \quad d_{12}=\dfrac{ac}{b^2\sigma _z^2}, \quad\text { and }\quad d_{22}=\dfrac{b^2\sigma _z^2+c^2\sigma _u^2}{b^2\sigma _z^2\sigma _u^2}. \end{aligned}$$

Therefore, the joint conditional PDF of $Z_0$ and U given $Y=y$ is reduced to the PDF of the bivariate truncated normal distribution

$$\begin{aligned} f(z_0, u\mid Y=y)=C\exp \left[ -\frac{1}{2}\left( z_0-\tau _1, u-\tau _2\right) \Sigma ^{-1}\left( z_0-\tau _1, u-\tau _2\right) ^\top \right] , \quad z_0, u>0, \end{aligned}$$

where C is the normalized constant. The desired result follows. $\square$