## 2 Properties of the LBS Distribution
### Fundamental Properties  
If a non-negative random variable X follows the GBS distribution established on an elliptically contoured kernel; say, \(G(\cdot)\) with shape parameter \(\alpha>0\), scale parameter \(\beta>0\), and a location parameter \(\gamma\), then the associated cumulative distribution function (CDF) is:  
\[F(x;\alpha,\beta,\gamma)=G\Bigg{(}\frac{1}{\alpha}\Bigg{[}\sqrt{\frac{x-\gamma }{\beta}}-\sqrt{\frac{\beta}{x-\gamma}}\Bigg{]}\Bigg{)},\quad x>\gamma\geq 0, \;\alpha,\beta>0. \tag{1}\]  
Now, assuming \(\gamma=0\), we can substitute the CDF of the logistic distribution \((1+e^{[-z]})^{-1}\) for the kernel \(G(\cdot)\) in (1), then the CDF of a non-negative random variable \(X\) represented as:  
\[F(x;\alpha,\beta)=\frac{1}{1+e^{-\frac{1}{\alpha}\Big{(}\sqrt{\frac{x}{\beta} }-\sqrt{\frac{\beta}{x}}\Big{)}}}, \tag{2}\]  
Evidently, if \(X\sim\text{LBS}(\alpha,\beta)\) the quantile function is then as follows:  
\[F^{-1}(u,\alpha,\beta)=\frac{\beta}{4}\bigg{(}\alpha\text{q}^{-1}(u)+\sqrt{[ \alpha\text{q}^{-1}(u)]^{2}+4}\bigg{)}^{2},\qquad 0<u<1. \tag{4}\]  
Clearly, using the inverse probability integral transform on the CDF (2), Equation (4) can be used to construct random variables having \(\text{LBS}(\alpha,\beta)\) as its distribution, where \(\text{q}^{-1}(u)\) is the inverse of the CDF of the standard logistic distribution. Accordingly, the median will be equal to \(\beta\).  
Upon using Equations (2) and (3), the HF can be obtained, which is defined as:  
\[h(x;\alpha,\beta)=\frac{1}{2\alpha x}\bigg{(}\sqrt{\frac{x}{\beta}}+\sqrt{ \frac{\beta}{x}}\bigg{)}\frac{1}{1+e^{-\frac{1}{4}\left(\sqrt{\frac{x}{\beta}} -\sqrt{\frac{\beta}{x}}\right)}}. \tag{5}\]  
The HF in (5) is unimodal when \(\alpha\to 0\) and decreases when \(\alpha\to\infty\).  
Figure 4: The HF for the LBS distribution with different values for \(\beta\) and \(\alpha=1\).  
Figure 3: The HF for the LBS distribution with different values for \(\alpha\) and \(\beta=1\).  
**Lemma 1**.: _To study the shape of HF, let us consider the following:_  
_a. The HF is decreasing for all values of \(x\) as \(\alpha\to\infty\)._  
_b. If \(\alpha\to\infty\), then the HF is an upside down function with critical point; say \(c_{\alpha,\beta}\), such that \(c_{\alpha,\beta}\geq(<)\beta\) when \(\alpha\leq(>)0.5\). For the proof of Lemma 1, see [15]._  
In the reliability analysis, the reliability (survival) function and the mean residual life (MRL) are significant characteristics of a lifetime model. On this basis and using the CDF we can obtain the reliability (survival) function as shown below:  
\[R(x;\alpha,\beta)=P(X\geq x)=\frac{e^{-\frac{1}{2}\left(\sqrt{\frac{\pi}{\beta }}-\sqrt{\frac{\beta}{x}}\right)}}{1+e^{-\frac{1}{2}\left(\sqrt{\frac{\pi}{ \beta}}-\sqrt{\frac{\beta}{x}}\right)}}, \tag{6}\]  
and  
\[m(t;\alpha,\beta)=E(X-t|X>t)=\int_{\mathbb{R}}exp\bigg{(}-\int_{t}^{t+x}h( \tau;\alpha,\beta)d\tau\bigg{)}dx, \tag{7}\]  
respectively, where \(h(\tau;\mu,\sigma)\) is the considered distribution's HF; see [16; 17] in this regard. Hence, it can readily conclude from the MRL expression that it has an opposite attitude to the HF.