(b) Let {X₁, X2,..., Xn] be a random sample from the probability distribution with probability density function: f(x; 0) = 8-1 for 20 0 is an unknown parameter. i. Derive the method of moments estimator of 8. 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 @ in part. i. is 02/(27n), check whether the estimator is a consistent estimator of 8.
(b) Let {X₁, X2,..., Xn] be a random sample from the probability distribution with probability density function: f(x; 0) = 8-1 for 20 0 is an unknown parameter. i. Derive the method of moments estimator of 8. 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 @ in part. i. is 02/(27n), check whether the estimator is a consistent estimator of 8.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![(b) Let (X₁, X2,..., Xn] be a random sample from the probability distribution
with probability density function:
Jo-¹ for 20 < x < -0
BIH
f(x; 0) =
otherwise
where > 0 is an unknown parameter.
i. Derive the method of moments estimator of 8.
ii. Is the estimator of 9 derived in part i. biased or unbiased? Justify your
answer.
Hint: You may use the fact that E(X) = E(X).
t
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 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5ed4928-4b60-4e32-a15e-9a625d0ebfdf%2Fdef9b95b-28bb-4434-b8df-e0986e85190f%2Fqn4kzxi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Let (X₁, X2,..., Xn] be a random sample from the probability distribution
with probability density function:
Jo-¹ for 20 < x < -0
BIH
f(x; 0) =
otherwise
where > 0 is an unknown parameter.
i. Derive the method of moments estimator of 8.
ii. Is the estimator of 9 derived in part i. biased or unbiased? Justify your
answer.
Hint: You may use the fact that E(X) = E(X).
t
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 0.
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