**6.6 Robust Bayesian estimation for survival function for binomial distribution**

From Eq. (69) and by using the squared loss function, we get a Bayesian estimator for the survival function, as follows [15]*:*

$$
\hat{\mathbf{S}}(\mathbf{x}) = \int\_0^1 \mathbf{S}(\mathbf{x}) \mathbf{f}(\mathbf{p}/\mathbf{s}) d\mathbf{p}
$$

$$
\hat{\mathbf{S}}\_{\mathrm{rob}}(\mathbf{x}) = \frac{1}{\beta(\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}}, \mathbf{n}^{\mathrm{m}} (1 - \mathbf{y}^{\mathrm{m}}) \sum\_{j=\mathrm{x}}^n \frac{\mathrm{n!}}{j!(\mathrm{n} - j)!} \prod\_{0}^1 \mathbf{p}^j (1 - \mathbf{p})^{\mathrm{n-j}} \mathbf{p}^{\mathrm{m} \cdot \mathbf{y}^{\mathrm{m}} - 1} (1 - \mathbf{p})^{\mathrm{n}^{\mathrm{m}} (1 - \mathbf{y}^{\mathrm{m}}) - 1} \mathrm{d}\mathbf{p}
$$

$$
\hat{\mathbf{S}}\_{\mathrm{rob}}(\mathbf{x}) = \frac{1}{\beta(\mathbf{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}}, \mathbf{n}^{\mathrm{m}} (1 - \mathbf{y}^{\mathrm{m}}) \sum\_{j=\mathrm{x}}^n \frac{\mathrm{n!}}{j!(\mathrm{n} - j)!} \prod\_{0}^1 \mathbf{p}^{\mathrm{n}^{\mathrm{m}} \mathbf{y}^{\mathrm{m}} + j - 1} (1 - \mathbf{p})^{\mathrm{n}^{\mathrm{m}} (1 - \mathbf{y}^{\mathrm{m}}) + \mathrm{n} - j - 1} \mathrm{d}\mathbf{p}
$$

Multiply and divide the equation by:

<sup>β</sup> <sup>n</sup>mym <sup>þ</sup> j, n<sup>m</sup> <sup>1</sup> � ym � � <sup>þ</sup> <sup>n</sup> � <sup>j</sup> � � ^ Srobð Þ¼ <sup>x</sup> <sup>1</sup> <sup>β</sup> nmym, nm <sup>1</sup> � ym � � �X<sup>n</sup> j¼x n! j!ð Þ n � j ! <sup>β</sup> <sup>n</sup>mym <sup>þ</sup> j, nm <sup>1</sup> � <sup>y</sup><sup>m</sup> � � <sup>þ</sup> <sup>n</sup> � <sup>j</sup> � � <sup>β</sup> nmym <sup>þ</sup> j, nm <sup>1</sup> � ym � � <sup>þ</sup> <sup>n</sup> � <sup>j</sup> � � <sup>ð</sup><sup>1</sup> 0 pnmymþj�<sup>1</sup> ð Þ <sup>1</sup> � <sup>p</sup> <sup>n</sup><sup>m</sup> <sup>1</sup>�y<sup>m</sup> ð Þþn�j�<sup>1</sup> dp ^ SRobð Þ¼ <sup>x</sup> <sup>1</sup> <sup>β</sup> nmym, nm <sup>1</sup> � ym � � � � <sup>X</sup><sup>n</sup> j¼x n! j!ð Þ n � j ! <sup>β</sup> nmym <sup>þ</sup> j, n<sup>m</sup> <sup>1</sup> � ym � � <sup>þ</sup> <sup>n</sup> � <sup>j</sup> � � (71)
