**5. Confidence Intervals**

In this section, the parameters' confidence intervals (CIs) are computed. Because our point estimate is the most likely value for the parameter, we should build the confidence intervals on it. CIs are a set of values (intervals) that serve as good approximations of an unknown population parameter. In this investigation, two types of CIs were computed.

### *5.1. Approximate Confidence Intervals*

*∂*β23

β3 Because the APE distribution's PDF is not symmetric, asymptotic CIs based on normality do not perform well. The underlying distribution is assumed to be APE. As a result, we believe that the parametric bootstrap percentile interval is preferable to the nonparametric one. Furthermore, it is well known that the nonparametric bootstrap percentile interval does not perform well in general. See Section 5.3.1 of Ref. [45] for more information. The parametric bootstrap interval with normal approximation or Studentization can be used. However, because this CI is symmetric, it may not be suitable for our asymmetric instance. According to large sample theory, the MLE results are consistent and regularly distributed, subject to certain regularity restrictions. According to large sample theory, the MLE results are consistent and regularly distributed, subject to certain regularity restrictions. Because

parameter MLE values are not in closed form, correct CIs cannot be obtained; instead, asymptotic CIs based on the asymptotic normal distribution of MLE values are computed.

Assume that ϕ = (β1,β2,β3, <sup>R</sup>). 1βˆ1 − β<sup>1</sup> , βˆ2 − β<sup>2</sup> , βˆ3 − β<sup>3</sup> , Rˆ − R \* is known to yield the asymptotic distribution of MLE values of N(0, <sup>σ</sup>), where σ = <sup>σ</sup>ij, i, j = 1, 2, 3, is the variance–covariance matrix of the unknown parameters. As previously established, the inverse of the Fisher information matrix is an estimator of the asymptomatic variance–covariance matrix.

The approximate 100(1 − ω)% two-sided CIs for ϕ are provided by

$$(\phi\_{\rm iL}, \phi\_{\rm iU}) : \phi\_{\rm i} \mp \mathbf{z}\_{1-\frac{\omega}{2}} \sqrt{\mathfrak{d}\_{\rm ij}}, \mathbf{i} = 1, 2, 3, 4. \tag{19}$$

where z1− ω2is the 1001 − ω2 -th upper percentile of the standard normal distribution.
