*3.1. Simulation Study*

We used the free software R version 3.4.4 [22] to carry out the Monte Carlo simulation study; the number of replications was 10,000. The pseudo-random samples were generated via Von Neumann's acceptance-rejection method [23]. This simple procedure requires the corresponding pdf *y* = *f*(*x*), a minorant and a majorant for *x* and a majorant for *y*; it is not necessary to implement the quantile function in this case. Four sample sizes, namely *n* = 50, 100, 200 and 500, and five different values for the vector of parameters were considered. For each scenario, we calculated the bias and the mean squared error (MSE) as follows:

$$\text{Bias}\_{i} = \frac{1}{10000} \sum\_{j=1}^{10000} \left( \hat{\xi}\_{ij} - \hat{\xi}\_{i} \right) \quad \quad \text{MSE}\_{i} = \frac{1}{10000} \sum\_{j=1}^{10000} \left( \hat{\xi}\_{ij} - \hat{\xi}\_{i} \right)^{2}$$

where *ξi* is the *i*-th element of the vector of parameters *ξ* = (*ξ*1, ... , *ξr*) and *ξij* is the estimate for *ξi* at the *j*-th replication. The log-likelihood function was maximized using the technique of simulated annealing, available by the optim subroutine, for which the user has to pass a vector *ξ*0 of initial values. At first, we took *ξ*0 = **1***<sup>r</sup>*, namely a *r* × 1 vector of ones, then we run one single replication considering sample size *n* = 50; the obtained estimates from this procedure were assigned to *ξ*0 and used in all of the aforementioned scenarios.

The results for both parameters of the Normal-Weibull density (8), shown in Table 1, indicate that the estimates are fairly close to the actual values. Moreover, as it would be expected, the bigger the sample size, the smaller the MSEs.


**Table 1.** Bias and MSE of the estimates under the maximum likelihood method for the Normal-Weibull model.

The results given in Table 2 sugges<sup>t</sup> that the estimates of the parameters of the Normal-log-logistic model (10) have similar behavior of those shown in Table 1, that is to say, the biases are quite small and the MSE decreases as the sample size increases.


**Table 2.** Bias and MSE of the estimates under the maximum likelihood method for the Normal-log-logistic model.
