**6.2 Bayesian estimation for survival function for binomial distribution**

Through Eq. (44) and by using the squared loss function we get a Bayes estimator for the survival function as follows [11] (**Figures 1** and **2**):

$$\hat{\mathbf{S}}(\mathbf{t}) = \int \mathbf{S}(\mathbf{t}) \mathbf{f}(\mathbf{p}/\mathbf{s}) d\mathbf{p}$$

$$\mathbf{f}(\mathbf{p}/\mathbf{s}) = \frac{1}{\beta(\alpha + \mathbf{s}, \mathbf{n} + \beta - \mathbf{s})} \mathbf{p}^{\alpha + \mathbf{s} - 1} (\mathbf{1} - \mathbf{p})^{\mathbf{n} + \beta - \mathbf{s} - 1}$$

$$\mathbf{S}(\mathbf{t}) = \sum\_{\mathbf{j} = \mathbf{t}}^{\mathbf{n}} \binom{n}{\mathbf{j}} \mathbf{p}^{\mathbf{j}} (\mathbf{1} - \mathbf{p})^{\mathbf{n} - \mathbf{j}}$$

$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{1}{\beta(\mathbf{a} + \mathbf{s}, \mathbf{n} + \beta - \mathbf{s})} \sum\_{j=t}^{\mathbf{n}} \frac{\mathbf{n}!}{j!(\mathbf{n} - \mathbf{j})!} \Bigg[ \mathbf{p}^{\mathbf{a} + s + j - 1} (\mathbf{1} - \mathbf{p})^{\mathbf{n} + \beta - s - 1 + \mathbf{n} - j} \mathbf{dp} \Bigg]$$

$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{\mathbf{1}}{\beta(\mathbf{a} + \mathbf{s}, \mathbf{n} + \beta - \mathbf{s})} \sum\_{j=t}^{\mathbf{n}} \frac{\mathbf{n}!}{j!(\mathbf{n} - j)!} \int\_{0}^{1} \mathbf{p}^{\mathbf{a} + s + j - 1} (\mathbf{1} - \mathbf{p})^{2\mathbf{n} + \beta - s - j - 1} \mathbf{dp}$$

**Figure 1.**

*It shows the behavior of the survival function by using the robust Bayes estimator, which is decreasing as the value of (t) increases, and this is consistent with the statistical theory [11].*

#### **Figure 2.**

*It shows the behavior of the survival function of a binomial distribution which is decreasing and this is consistent with the statistical theory [13].*

$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{\mathbf{1}}{\beta(\alpha + \mathbf{s}, \mathbf{n} + \beta - \mathbf{s})} \sum\_{j=t}^{\mathbf{n}} \frac{\mathbf{n}!}{j!(\mathbf{n} - j)!} \Bigg\} \overset{\mathbf{1}}{\mathbf{p}}^{\mathbf{n} + s + j - 1} (\mathbf{1} - \mathbf{p})^{2\mathbf{n} + \beta - s - j - 1} \mathbf{dp}$$

Multiply and divide by:

$$\mathfrak{f}(\mathfrak{a} + \mathbf{s} + \mathbf{j}, 2\mathbf{n} + \mathfrak{f} - \mathbf{s} - \mathbf{j})$$

$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{1}{\mathfrak{f}(\mathfrak{a} + \mathbf{s}, \mathbf{n} + \mathfrak{f} - \mathbf{s})} \sum\_{j=t}^{n} \frac{\mathfrak{n}!}{\mathfrak{j}!(\mathfrak{n} - \mathfrak{j})!} \frac{\mathfrak{f}(\mathfrak{a} + \mathbf{s} + \mathbf{j}, 2\mathbf{n} + \mathfrak{f} - \mathbf{s} - \mathbf{j})}{\mathfrak{f}(\mathfrak{a} + \mathbf{s} + \mathbf{j}, 2\mathbf{n} + \mathfrak{f} - \mathbf{s} - \mathbf{j})} \prod\_{0}^{1} \mathsf{P}^{\mathfrak{a} + \mathbf{s} + \mathfrak{j} - 1}(\mathbf{1})$$

$$- \mathbf{p} \big)^{2\mathfrak{n} + \mathfrak{P} - \mathbf{s} - \mathfrak{j} - 1} \mathrm{d}\mathbf{p}$$

$$\hat{\mathbf{S}}(\mathbf{t}) = \frac{1}{\mathfrak{f}(\mathfrak{a} + \mathbf{s}, \mathbf{n} + \mathfrak{f} - \mathbf{s})} \sum\_{j=t}^{n} \frac{\mathfrak{n}!}{\mathfrak{j}!(\mathfrak{n} - \mathfrak{j})!} \mathfrak{f}(\mathfrak{a} + \mathbf{s} + \mathbf{j}, 2\mathbf{n} + \mathfrak{f} - \mathbf{s} - \mathbf{j}) \tag{46}$$
