**3.7 Robust Bayesian estimation for survival function for Weibull distribution**

From Eq. (37) and by using the squared loss function we get a Bayes estimator for the survival function as follows [12]:

$$
\hat{\mathbf{S}}(\mathbf{t}) = \int\_0^\infty \mathbf{S}(\mathbf{t}) \, \mathbf{f}(\mathbf{0}/\mathbf{t}) \, \mathbf{d}\theta
$$

$$
\hat{\mathbf{S}}(\mathbf{t}) = \frac{\left(\mathbf{n}^\mathrm{m} \mathbf{y}^\mathrm{m}\right)^{\mathrm{n}^\mathrm{m}+1}}{\mathrm{r}(\mathbf{n}^\mathrm{m}+1)} \int\_0^\infty \mathbf{e}^{-\frac{\theta}{\theta}} \theta^{-(\mathbf{n}^\mathrm{m}+1)-1} \mathbf{e}^{-\frac{\mathbf{n}^\mathrm{m} \mathbf{y}^\mathrm{m}}{\theta}} \, \mathbf{d}\theta
$$

$$
\hat{\mathbf{S}}(\mathbf{t}) = \frac{\left(\mathbf{n}^\mathrm{m} \mathbf{y}^\mathrm{m}\right)^{\mathrm{n}^\mathrm{m}+1}}{\mathrm{r}(\mathbf{n}^\mathrm{m}+1)} \int\_0^\infty \theta^{-(\mathbf{n}^\mathrm{m}+1)-1} \mathbf{e}^{-\frac{\left(\mathbf{n}^\mathrm{m} \mathbf{y}^\mathrm{m} + \theta\right)}{\mathbf{0}}} \, \mathbf{d}\theta
$$

By using the transformation:

$$\text{let } \mathbf{z} = \frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}} + \mathbf{t}^{\boldsymbol{\theta}}\right)}{\boldsymbol{\Theta}} \Longrightarrow \boldsymbol{\Theta} = \frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}} + \mathbf{t}^{\boldsymbol{\theta}}\right)}{\mathbf{z}}; \left|\mathbf{J}\right| = \left|-\frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}} + \mathbf{t}^{\boldsymbol{\theta}}\right)}{\mathbf{z}^2}\right| $$
 
$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}}\right)^{\mathbf{n}^{\mathbf{m}} + 1}}{\mathbf{r}(\mathbf{n}^{\mathbf{m}} + 1)} \int\_{0}^{\mathbf{e}} \left(\frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}} + \mathbf{t}^{\boldsymbol{\theta}}\right)}{\mathbf{z}}\right)^{-(\mathbf{n}^{\mathbf{m}} + 1) - 1} \mathbf{e}^{-\mathbf{z}} \frac{\left(\mathbf{n}^{\mathbf{m}} \mathbf{y}^{\mathbf{m}} + \mathbf{t}^{\boldsymbol{\theta}}\right)}{\mathbf{z}^2} \,\mathrm{d}\mathbf{z} $$

$$
\hat{\mathbf{S}}(\mathbf{t}) = \frac{\left(\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}}\right)^{\mathrm{n}^{\mathrm{m}} + 1}}{\mathrm{r}(\mathbf{n}^{\mathrm{m}} + \mathbf{1})} \left(\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}} + \mathbf{t}^{\emptyset}\right)^{-(\mathbf{n}^{\mathrm{m}} + 1)} \begin{cases} (\mathbf{z})^{\mathrm{n}^{\mathrm{m}}} \mathbf{e}^{-\mathbf{z}} \,\mathrm{d}\mathbf{z} \\ \mathbf{0} \end{cases}
$$

$$
\hat{\mathbf{S}}\_{\mathrm{Rob}}(\mathbf{t}) = \left(\frac{\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}}}{\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}} + \mathbf{t}^{\emptyset}}\right)^{\mathrm{n}^{\mathrm{m}} + 1} \tag{39}
$$
