Suppose that we have observations Xij = Mi + tij, = = 1, 2, ..., m, j = 1, 2, ..., n, 2 where cij's are i.i.d._from N(0,1). We can see that the MLEs of parameters (M1, M2, ...., Mm) are (X1, X2, ..., X.). , Xm 1. Consider the following prior distribution where Mi i.i.d. ~ N(0,72) (we start with assuming that hyperparameters 0 and 72 are known constants.). Find the Bayes estimator of vector (M1, M2, Mom) under the squared error loss. .... 2. Evaluate the bias and variance of this estimator. 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 and 72. 4. We can get a new estimator of (M1, M2, plem), by replacing hyperparame- ters 6 and 2 with MLEs @ 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.) , Suppose that we have observations Xij = Mi + tij, = = 1, 2, ..., m, j = 1, 2, ..., n, 2 where cij's are i.i.d._from N(0,1). We can see that the MLEs of parameters (M1, M2, ...., Mm) are (X1, X2, ..., X.). , Xm 1. Consider the following prior distribution where Mi i.i.d. ~ N(0,72) (we start with assuming that hyperparameters 0 and 72 are known constants.). Find the Bayes estimator of vector (M1, M2, Mom) under the squared error loss. .... 2. Evaluate the bias and variance of this estimator. 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 and 72. 4. We can get a new estimator of (M1, M2, plem), by replacing hyperparame- ters 6 and 2 with MLEs @ 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.)