e-Statistics > Medical Signal Detection

Stein-like Estimator

We consider an empirical Bayes approach to a Stein's estimator by Efron and Morris, and use it as a guideline to construct a ``Stein-like'' estimator for $ \lambda_{ij}$ . Here we fix the adverse event $ j$ to obtain the estimate $ \lambda_{ij}$ simultaneously for all $ i$ . We assume that approximately $ N_{ij}/N_{i \cdot} \sim N(p_{ij}, D_{i})$ are independent for all $ i$ , and that $ p_{ij} \sim N(p_{\cdot j}, A)$ . Then we can define the estimator of $ p_{ij}$ by

$\displaystyle \hat{p}_{ij}^{ST} = p_{\cdot j}
+ (1 - B_i)\left(\frac{N_{ij}}{N_{i \cdot}} - p_{\cdot j}\right)
$

where $ B_i = D_i/(A + D_i)$ . Here we use the estimate

$\displaystyle \hat{D}_{i} = \frac{\hat{p}_{ij}(1 - \hat{p}_{ij})}{N_{i \cdot}}.
$

Furthermore, the estimate $ \hat{A}$ will be obtained as the MLE of $ A$ on the basis of $ S_i = (N_{ij}/N_{i \cdot})^2 \sim (A + D_i)\chi^2$ ; thus, it becomes the solution to

$\displaystyle \hat{A} = \sum_i (S_i - D_i) I_i(\hat{A})
\left/ \sum_k I_k(\hat{A}) \right.
$

with $ I_i(\hat{A}) = 1/[2(A + D_i)^2]$ . Together we can obtain the stein-like estimator of relative reprot rate by

$\displaystyle \lambda_{ij}^{ST} = \frac{\hat{p}_{ij}^{ST}}{p_{\cdot j}}
= 1 + (1 - B_i)\left(\lambda_{ij}^{RR} - 1\right)
$

where we use $ \lambda_{ij}^{RR} \approx N_{ij}/(p_{\cdot j} N_{i \cdot})$ .


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