Question

Let X1, ..., Xn be independent and identically distributed t-distributed random variables with k > 2 degrees of freedom.

(a) Find an estimator for k using the method of moments.
(b) In which cases can the method of moments not provide an estimator for k?
Note: The expected value of a t-distributed random variable with k > 2 degrees of freedom is 0 and its variance is k/k-2.

Answer 1

To find an estimator for k using the method of moments, we equate the sample moments with the theoretical moments. An estimator for k is n. The method of moments may not provide an estimator for k when the equations are not solvable.

Step-by-step explanation:

(a) To find an estimator for k using the method of moments, we can equate the sample moments with the theoretical moments. For the t-distributed random variables, the expected value is 0 and the variance is k/(k-2). So, equating the sample mean to 0, we can solve for k:

Mean of X = 0

Sum of X_i / n = 0

Sum of X_i = 0

Since the variables are identically distributed, the sum of the variables will be 0. So, n * E(X_i) = 0.

Since E(X_i) = 0, we have n * 0 = 0.

Thus, n = 0 or k.

Therefore, an estimator for k using the method of moments is n.

(b) The method of moments may not provide an estimator for k in cases where the sample moments cannot be equated with the theoretical moments. This can happen when there are not enough equations or when the equations are not solvable.