Question

Suppose that $X_1, X_2, ..., X_n$ is a random sample coming from the following probability mass function: $f(x, r, p) = \binom{x-1}{r-1} p^r (1-p)^{x-r}$ for $x = r, r+1, r+2,...$ i. (6 pts) Find the method of moments estimator of the parameter $p$ above. Assume that $r$ is known. Hint: $E[X] = r/p$ ii. (6 pts) Find the maximum likelihood estimator of the parameter $p$ above. Assume that $r$ is already known. iii. (3 pts) Find the standard error of the method of moment estimator in part (i). iv. (4 pts) Find MSE of the maximum likelihood estimator you found in part (ii). Hint: $V[X] = r(1-p)/p^2$ v. (3 pts) Find the maximum likelihood estimator of $1/p$.

          Suppose that $X_1, X_2, ..., X_n$ is a random sample coming from the following probability mass function:

$f(x, r, p) = \binom{x-1}{r-1} p^r (1-p)^{x-r}$ for $x = r, r+1, r+2,...$

i. (6 pts) Find the method of moments estimator of the parameter $p$ above.
Assume that $r$ is known.
Hint: $E[X] = r/p$
ii. (6 pts) Find the maximum likelihood estimator of the parameter $p$ above.
Assume that $r$ is already known.
iii. (3 pts) Find the standard error of the method of moment estimator in part (i).
iv. (4 pts) Find MSE of the maximum likelihood estimator you found in part (ii).
Hint: $V[X] = r(1-p)/p^2$
v. (3 pts) Find the maximum likelihood estimator of $1/p$.

Suppose that X1, X2, ..., Xn is a random sample coming from the following probability mass function:

f(x, r, p) = x-1r-1 p^r (1-p)^x-r for x = r, r+1, r+2,...

i. (6 pts) Find the method of moments estimator of the parameter p above.
Assume that r is known.
Hint: E[X] = r/p
ii. (6 pts) Find the maximum likelihood estimator of the parameter p above.
Assume that r is already known.
iii. (3 pts) Find the standard error of the method of moment estimator in part (i).
iv. (4 pts) Find MSE of the maximum likelihood estimator you found in part (ii).
Hint: V[X] = r(1-p)/p^2
v. (3 pts) Find the maximum likelihood estimator of 1/p.

Added by Juan J.

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