- Suppose That We Have Observations Xij Mi Tij I 1 2 M J 1 2 N Where Tij S Are I I D From N 0 1 We 1 (63.31 KiB) Viewed 38 times
PLEASE.
I WILL RATE POSITIVE.
Suppose that we have observations Xij = Mi + tij, i=1,2,..., m, j = 1, 2, ..., n, where tij's are i.i.d._from N(0,1). We can see that the MLEs of parameters (141, H2, ...., Hlm) are (X1, X2, ..., Xm). 1. 1 . 2. 3. In the questions above, we see that the bias of this estimator depends on the choice of hyperparameters. To eliminate unnecessary bias, in practice we may want to estimate them directly from data. Find the MLEs o and 12. 4. We can get a new estimator of (H1, H2, ..., fem), by replacing hyperparame- ters 6 and 72 with MLEs Ộ and 12 in the existing Bayes estimator above. In literature, this is called empirical Bayes estimator. Compare the risks between the empirical Bayes estimator and MLE (X1, X2, ..., Xm) using a numerical simulation approach. (Hint: Consider scenarios such as small n large m, and small m large n.)