Bias and MSE
We estimate the unknown mean θ of a random variable X with unit variance by forming the sample mean Mn=(X1+⋯+Xn)/n of n i.i.d. samples Xi and then forming the estimator
Θˆn=13⋅Mn.
Your answers below can be functions of θ and n. Follow standard notation and use 'theta' to indicate θ.
The bias E[Θˆn]−θ of this estimator is:
unanswered
The mean squared error of this estimator is:
unanswered
For biais: -2*theta/3
(1/(9*n))+(4*theta^2)/9
bias: -2*theta/3
mse: (1/(9*n))+(4*theta^2)/9
-2*theta/3
-2*theta/3
To determine the bias and mean squared error (MSE) of the given estimator, we need to compute the expected value and variance of the estimator.
First, let's find the bias E[Θˆn]−θ of the estimator Θˆn.
The estimator Θˆn is defined as Θˆn = (1/3)⋅Mn, where Mn is the sample mean of n i.i.d. samples Xi.
To compute the bias, we need to calculate the expected value of the estimator E[Θˆn].
E[Θˆn] = E[(1/3)⋅Mn]
Since Mn is the sample mean, we know that E[Mn] = θ, as it is an unbiased estimator of θ.
Therefore, we have:
E[Θˆn] = E[(1/3)⋅Mn] = (1/3)⋅E[Mn] = (1/3)⋅θ
Hence, the bias E[Θˆn]−θ of this estimator is given by:
Bias = E[Θˆn] - θ = (1/3)⋅θ - θ = -2/3⋅θ.
Now, let's determine the mean squared error (MSE) of the estimator.
The MSE is defined as the expected value of the squared difference between the estimator and the true value:
MSE = E[(Θˆn - θ)^2]
Substituting Θˆn = (1/3)⋅Mn, we have:
MSE = E[((1/3)⋅Mn - θ)^2]
Expanding the squared term, we get:
MSE = E[(1/9)⋅(Mn^2 - 2⋅Mn⋅θ + θ^2)]
Since Mn is the sample mean, we know that Var(Mn) = 1/n (as each Xi has a unit variance).
We can rewrite the MSE as follows:
MSE = (1/9)⋅(E[Mn^2] - 2⋅E[Mn⋅θ] + θ^2)
To compute the MSE, we need to find the expectations E[Mn^2] and E[Mn⋅θ].
Since the Xi's are identically distributed, we have:
E[Mn^2] = Var(Mn) + [E(Mn)]^2 = 1/n + θ^2
E[Mn⋅θ] = θ⋅E[Mn] = θ⋅θ = θ^2
Substituting these values back into the equation for MSE, we get:
MSE = (1/9)⋅(1/n + θ^2 - 2⋅θ^2 + θ^2) = (1/9)⋅(1/n - θ^2)
Therefore, the mean squared error (MSE) of this estimator is:
MSE = (1/9)⋅(1/n - θ^2)
mse: 4*theta^2/9
Helper's answer is kinda right, but input it like:
(4*theta^2)/9