Second-order skewness of the maximum likelihood estimators in symmetric nonlinear regressions

The symmetric nonlinear regression model encompasses a wide class of models like the nonlinear normal, Student-t, and power exponential regression models, among others. We provide a simple closed-form expression, in matrix form, for the second-order skewness of the maximum likelihood estimators of the regression parameters, as well as a simple expression for the second-order skewness of the maximum likelihood estimator of the scale parameter in the symmetric nonlinear regression model. The matrix formula involves simple operations on matrices and vectors, which is quite suitable for computer implementation. We present Monte Carlo simulations to verify the behaviour of the second-order skewness in small-sized samples in this class of regression models. Real data applications are also considered to show the practical importance of our results.