Let \sigma =1 and consider the special case of only two observations (n=2). Write down a formula for the mean squared error \mathbb {E}[(\hat{\Theta }_1-\Theta _1)^2], as a function of t_1 and t_2. Enter t_1 for t_1 and t_2 for t_2.
\mathbb {E}[(\hat{\Theta }_1-\Theta _1)^2]=\quad
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Consider the "experimental design" problem of choosing when to make measurements. Under the assumptions of the previous part, and under the constraints 0\leq t_1,t_2 \leq 10, find the values of t_1 and t_2 that minimize the mean squared error associated with the MAP estimator.
t_1=\quad
unanswered
t_2=\quad
unanswered
To find the formula for the mean squared error, let's start by assuming that the observations are independent and identically distributed with a mean of \Theta_1 and a variance of \sigma^2.
The estimator \hat{\Theta}_1 is a linear combination of the observations, given by \hat{\Theta}_1 = t_1 X_1 + t_2 X_2, where X_1 and X_2 are the two observations.
The mean squared error, \mathbb{E}[(\hat{\Theta}_1 - \Theta_1)^2], can be calculated as follows:
\mathbb{E}[(\hat{\Theta}_1 - \Theta_1)^2] = \mathbb{E}[(t_1 X_1 + t_2 X_2 - \Theta_1)^2]
= \mathbb{E}[t_1^2 X_1^2 + t_2^2 X_2^2 + 2t_1 t_2 X_1 X_2 - 2t_1 X_1 \Theta_1 - 2t_2 X_2 \Theta_1 + \Theta_1^2]
= t_1^2 \mathbb{E}[X_1^2] + t_2^2 \mathbb{E}[X_2^2] + 2t_1 t_2 \mathbb{E}[X_1 X_2] - 2t_1 \Theta_1 \mathbb{E}[X_1] - 2t_2 \Theta_1 \mathbb{E}[X_2] + \Theta_1^2
Since the observations are independent and identically distributed, we have:
\mathbb{E}[X_1^2] = \mathbb{E}[X_2^2] = \sigma^2
\mathbb{E}[X_1 X_2] = 0 (covariance of independent variables)
\mathbb{E}[X_1] = \mathbb{E}[X_2] = \Theta_1
Substituting these values into the mean squared error formula, we get:
\mathbb{E}[(\hat{\Theta}_1 - \Theta_1)^2] = t_1^2 \sigma^2 + t_2^2 \sigma^2 - 2t_1 \Theta_1^2 - 2t_2 \Theta_1^2 + \Theta_1^2
= \sigma^2 (t_1^2 + t_2^2) - 2 \Theta_1 (t_1 + t_2) + \Theta_1^2
In summary, the formula for the mean squared error is:
\mathbb{E}[(\hat{\Theta}_1 - \Theta_1)^2] = \sigma^2 (t_1^2 + t_2^2) - 2 \Theta_1 (t_1 + t_2) + \Theta_1^2
Now, let's proceed to find the values of t_1 and t_2 that minimize the mean squared error associated with the MAP estimator.