Let sigma=1 and consider the special case of only two observations (n=2). Write down a formula for the mean squared error E[(delta_1 hat-delta_1)^2] as a function of t_1 and t_2. Enter t_1 for t_1 and t_2 for t_2.
In the case of only two observations (n=2), the mean squared error can be calculated using the formula:
E[(delta_1 hat - delta_1)^2] = Var(delta_1 hat) + (Bias(delta_1 hat))^2
Since we are given that sigma=1, the variance, Var(delta_1 hat), can be calculated as:
Var(delta_1 hat) = sigma^2 / n = 1/2 = 0.5
Bias(delta_1 hat) represents the bias of the estimator and, in this case, will depend on the specific estimators used for delta_1 hat. Without further information, it is not possible to determine the exact value of Bias(delta_1 hat) and thus the mean squared error. Therefore, we cannot provide a specific formula in terms of t_1 and t_2 without additional information on the estimators employed.
To calculate the mean squared error (MSE) for the special case of only two observations (n=2), we need to determine the formula based on the given information.
Let's assume that delta_1 hat represents the estimate of delta_1, and sigma=1 (standard deviation).
The mean squared error (MSE) can be calculated using the following formula:
MSE = E[(delta_1 hat - delta_1)^2]
Given that there are only two observations, t_1 and t_2, we can calculate the estimate of delta_1 (delta_1 hat) as the mean of the two observations:
delta_1 hat = (t_1 + t_2) / 2
Now, we can substitute this estimate into the formula for MSE:
MSE = E[((t_1 + t_2) / 2 - delta_1)^2]
Therefore, the formula for the mean squared error (MSE) in terms of t_1 and t_2 is:
MSE = E[((t_1 + t_2) / 2 - delta_1)^2]