In Section 2, we briefly discuss the FSS distribution and FSTN distributions.
The above extension of (1) is referred to as flexible skew-t-normal (FSTN) distribution, denoted by FSTN([mu], [omega], [alpha], v).
It is noted that the FSTN distribution is proposed within the general framework of FSS distribution by combining with the definition of STN distribution and, as a consequence, it shares analogous feature with these two distributions.
Figure 1 displays the density functions of FSN and STN as well as FSTN distributions with four different situations considered, namely, [[alpha].sub.1] = 1, [[alpha].sub.3] = 0, and v = 10; [[alpha].sub.1] = [[alpha].sub.3] = 1 and v = 6; [[alpha].sub.1] = 1, [[alpha].sub.3] = -1, and v = 4; [[alpha].sub.1] = -1, [[alpha].sub.3] = 1, and v = 4, respectively, with [mu] = 1, [omega] = 1.5 for all cases.
To improve the efficiency of the algorithm and to facilitate statistical inference of the nonlinear models with FSTN distribution, we put forward the following profile likelihood method based on (3) and (6).
with two kinds of skewed distributions for random error as follows: Case (I): [epsilon] ~ STN([mu], [omega], [alpha], v) and Case (II): [epsilon] ~ FSTN ([mu], [lambda], [alpha], v).
Similar to previous analysis, each simulated data set is fitted under STN and FSN as well as FSTN scenarios using three different estimation algorithms.
To study the consistence properties of ML estimate, we focus on the situation that the true distribution for random error is Case (II) whereas the fitting distribution is FSTN too.
Tables 1 and 2 show that in general FSTN distribution enjoys more robustness and flexibility in modeling data with skewness and heavy tails as well as multimodality in comparison with other skewed alternatives and the implementation of MPNR method brings more accuracy and improvement for model estimation in the context of nonlinear regression with this new distribution.