An be a andom with probability density function: 8-1 for 20< I<-8 f(x; 0) = = otherwise where > 0 is an unknown parameter. i. Derive the method of moments estimator of 0. ii. Is the estimator of 0 derived in part i. biased or unbiased? Justify your answer. Hint: You may use the fact that E(X) = E(X). iii. Given that the variance of the method of moments estimator of 9 in part i. is 02/(27n), check whether the estimator is a consistent estimator of 0. (b) Let X₁, 2,. the

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(b) Let (X₁, X2,..., Xn} be a random sample from the probability distribution
with probability density function:
Se-¹ for 20< I<-0
f(x; 0) =
otherwise
where > 0 is an unknown parameter.
i. Derive the method of moments estimator of 0.
ii. Is the estimator of 6 derived in part i. biased or unbiased? Justify your
answer.
Hint: You may use the fact that E(X) = E(X).
iii. Given that the variance of the method of moments estimator of 0
in part i. is 02/(27n), check whether the estimator is a consistent
estimator of 6.
Transcribed Image Text:(b) Let (X₁, X2,..., Xn} be a random sample from the probability distribution with probability density function: Se-¹ for 20< I<-0 f(x; 0) = otherwise where > 0 is an unknown parameter. i. Derive the method of moments estimator of 0. ii. Is the estimator of 6 derived in part i. biased or unbiased? Justify your answer. Hint: You may use the fact that E(X) = E(X). iii. Given that the variance of the method of moments estimator of 0 in part i. is 02/(27n), check whether the estimator is a consistent estimator of 6.
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