*3.2. Simulation Study*

In this Section, we present a brief simulation study to assess the performance of MLE in the GSC model. To draw values from the model, we used the inversion method. If *U* ∼ *U*(0, <sup>1</sup>), then

$$Z = \mu + \sigma \sinh^{-1}\left(\lambda^{-1} \tan\left[\pi/2 - \pi \exp\left(-G^{-1}\left(lL;\phi\right)\right)\right]\right) \sim GSC(\phi, \lambda, \mu, \sigma),$$

where *<sup>G</sup>*−<sup>1</sup>(·; *φ*) is the inverse of the cdf of the gamma distribution with shape and rate parameters equal to *φ* and 1, respectively. In all scenarios, we considered *μ* = 0 and *σ* = 1, three values for *φ*—0.75, 1, and 1.5—and three values for *λ*—0.5, 1, and 2. We also considered three sample sizes: 100, 200, and 500. For each combination of the parameters, we drew 1,000 samples and computed the ML estimates. Table 2 summarizes the results considering the average of the bias (bias), the root of the estimated mean squared error (RMSE), and the 95% coverage probability (CP). Note that in all cases, the bias and RMSE decreased when the sample size was increased, suggesting that the estimators are consistent. Finally, we also remark that the coverage probability converged to the nominal values used for the construction of the confidence intervals when the sample size was increased, suggesting that the normality for the ML estimates is reasonable in sample sizes.

**Table 2.** Simulation study for the GSC(*φ*, *λ*, *μ* = 0, *σ* = 1) model.



**Table 2.** *Cont*.
