Question

Let XI, X2, Xn be a random sample from a N(μ, σ2) population. For any constant k 〉 0, define σ2-21-1(k-_. con- sider as an estimator for σ2 (a) Compute the bias of σ in terms of k. [Hint: The sample variance is unbiased, and σ (n-1)s2/k.] (b) Compute the variance of in terms of k. Hint: (c) Compute the mean squared error of k in terms of k. (d) For what values of k is the mean squared error of minimized?

Answer 1

(a) The bias of the estimator for σ^2, denoted as MSE(σ^2), is defined as the difference between the expected value of the estimator and the true value of the parameter. In this case, the estimator is (n-1)s^2 / k, where n is the sample size, s^2 is the sample variance, and k is a constant greater than 0.

The expected value of the sample variance s^2 is equal to the true variance σ^2, since s^2 is an unbiased estimator of σ^2. Therefore, the bias of the estimator for σ^2 is given by:

Bias(σ^2) = E[(n-1)s^2 / k] - σ^2

Now substituting the value of s^2, we get:

Bias(σ^2) = E[(n-1)(σ^2) / k] - σ^2

Simplifying further:

Bias(σ^2) = (n-1)σ^2 / k - σ^2

(b) The variance of the estimator for σ^2, denoted as Var(σ^2), is given by the variance of the sample variance s^2, which can be calculated as:

Var(σ^2) = Var[(n-1)s^2 / k]

Since s^2 is an unbiased estimator of σ^2, Var[(n-1)s^2] = 2(n-1)^2σ^4 / (k^2(n-3)), using the known formula for the variance of sample variance.

Therefore, substituting the value of Var[(n-1)s^2] into the equation, we get:

Var(σ^2) = 2(n-1)^2σ^4 / (k^2(n-3))

(c) The mean squared error (MSE) of the estimator for σ^2, denoted as MSE(σ^2), is the sum of the variance and the square of the bias:

MSE(σ^2) = Var(σ^2) + Bias(σ^2)^2

Substituting the values of Var(σ^2) and Bias(σ^2) from parts (a) and (b), respectively, we get:

MSE(σ^2) = 2(n-1)^2σ^4 / (k^2(n-3)) + [(n-1)σ^2 / k - σ^2]^2

(d) To minimize the mean squared error of the estimator, we need to find the value of k that minimizes the MSE(σ^2). This can be done by taking the derivative of the MSE(σ^2) with respect to k, setting it equal to zero, and solving for k. However, since the equation for MSE(σ^2) is quite complex, it may not have a simple closed-form solution for k. In practice, numerical optimization techniques can be used to find the value of k that minimizes the MSE(σ^2) by iterating over a range of possible values for k and calculating the corresponding MSE(σ^2) for each value. The value of k that gives the lowest MSE(σ^2) can then be chosen as the optimal value.