Bayes Estimator for Survival Function

Definition

The Bayes estimator of $S(t)$ with the Squared Error Loss and a Dirichlet process prior ${\mathcal{P}}_{\alpha }$ is given by ${\hat{S}}_{\alpha }(t)=\frac{\alpha (t,\infty )+N_y(t)}{\alpha (0,\infty )+n}{\prod }_{i:y_i\le t}{\left[\frac{\alpha [y_i,\infty )+(n-i+1)}{\alpha [y_i,\infty )+(n-i)}\right]}^{1-{\delta }_i}$ where the squared loss is $L(\hat{S},S)={\int }_0^{\infty }(\hat{S}(t)-S(t){)}^{2}dw(t)$ where $w$ is any non-negative non-decreasing function, the parameter $\alpha$ which is a finite non-negative measure on $(0,\infty )$, and $N_y(t)=\sum _{i=1}^{n}I(Y_i>t)=\#(Y_i>t)$.

Examples

If $\frac{\alpha (t,\infty )}{\alpha (0,\infty )}=\exp [-{\lambda }_{0}t]$, then In many cases, $\mathrm{MLE}[{\hat{S}}_{\alpha }(t)]\le \mathrm{MLE}[\hat{S}(t)]$