Consider the family of distributions with pmf pX(x) = p if x = −1, 2p if...

Consider the family of distributions with pmf pX(x) = p if x = −1, 2p if x = 0, 1 − 3p if x = 1 . Here p is an unknown parameter, and 0 ≤ p ≤ 1/3. Let X1, X2, . . . , Xn be iid with common pmf a member of this family.

(i) Find the MOM estimator of p.

(ii) Find the MLE estimator of p. (Consider the statistics A = the number of i with Xi = −1, B = the number of i with Xi = 0, C = the number of i with Xi = 1.)

(iii) A sample from this distribution produced the values { 1, 1, 1, -1, 0, 0, 1, -1, 1, 1, -1, 1, 1, -1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 1, -1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, -1, 0, -1, 1, 1, 0, 0 } (There are 8 -1’s, 15 0’s and 27 1’s.) What are the MOM and MLE estimates of p for this data set? Which seems to fit the data better? Explain your answer.

Expert Solution

orchestra answered 2 years ago

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