2.1.2. The Normal-Log-Logistic Distribution

The Log-logistic distribution is commonly applied to reliability and oftentimes it works well as a lifetime model. Its cdf is given by *GLL*(*x*|*<sup>α</sup>*, *β*) = 1 − -1 + (*x*/*α*)*<sup>β</sup>*−<sup>1</sup> for *x* ≥ 0, where *α*, *β* > 0. The Normal-log-logistic cdf is easily obtained replacing the parent distribution *G* in Equation (4) by *GLL*. Thus:

$$F\_{NLL}(\mathbf{x}) = \Phi\left[\left(\frac{\mathbf{x}}{\mathfrak{a}}\right)^{\mathfrak{f}} - \left(\frac{\mathbf{x}}{\mathfrak{a}}\right)^{-\mathfrak{f}}\right],\tag{9}$$

for *x* ≥ 0. Taking the derivative of Equation (9) with respect to *x*, we ge<sup>t</sup> to the pdf:

$$f\_{NLL}(\mathbf{x}) = \phi \left[ \left( \frac{\mathbf{x}}{a} \right)^{\beta} - \left( \frac{\mathbf{x}}{a} \right)^{-\beta} \right] \left[ 1 + \left( \frac{\mathbf{x}}{a} \right)^{2\beta} \right] \beta a^{\beta} \mathbf{x}^{-\beta -1}. \tag{10}$$

Figure 3 shows plots of pdf and hrf for different values of *α* and *β*. It is worth noting that the Normal-log-logistic distribution may have a decreasing hrf of early failure. It is also possible for the hrf to be increasing or unimodal.

**Figure 3.** Plots of pdf and hrf for the Normal-log-logistic distribution.

Pearson's moment coefficient of skewness for the Normal-log-logistic distribution is depicted in Figure 4.

**Figure 4.** Skewness of the Normal-log-logistic distribution.
