iid (4) Let x1, X2, ... ,Xn Pareto with the following density ( 2021x³, for x > 0, føx) = otherwise, where o e (0, co). Find an ML estimator of e if it exists. The following ¨answers" have been proposed. (a) The likelihood function of o is 2"o2n/ X;. This is an even degree convex monomial defined over the whole real line. Hence, it only has a minimum but no maximum. Hence, no MLE can exist. (b) The likelihood function of e is 2"02n/ [I?=1 X¡. This is a polynomial that keeps increasing as o gets large over the positive half of the real line. Hence, no MLE can exist. (c) The likelihood function is a function of e over the interval (0, x(1)] where x(1) = min(X1, X2, . , Xn) and the likelihood function equals zero outside this interval. Since the left end point of the interval is not included, the maximum may not exist. Therefore, the MLE may not exist. (d) The likelihood function is a strictly increasing function of e over the interval (0, x(1) where x(1) = min(X1, X2, … Xn) and the likelihood function equals zero outside this interval. Therefore, the MLE exists and equals e = X(1)- (e) None of the above. The correct answer is (a) (b) (c) (d) (e) N/A
iid (4) Let x1, X2, ... ,Xn Pareto with the following density ( 2021x³, for x > 0, føx) = otherwise, where o e (0, co). Find an ML estimator of e if it exists. The following ¨answers" have been proposed. (a) The likelihood function of o is 2"o2n/ X;. This is an even degree convex monomial defined over the whole real line. Hence, it only has a minimum but no maximum. Hence, no MLE can exist. (b) The likelihood function of e is 2"02n/ [I?=1 X¡. This is a polynomial that keeps increasing as o gets large over the positive half of the real line. Hence, no MLE can exist. (c) The likelihood function is a function of e over the interval (0, x(1)] where x(1) = min(X1, X2, . , Xn) and the likelihood function equals zero outside this interval. Since the left end point of the interval is not included, the maximum may not exist. Therefore, the MLE may not exist. (d) The likelihood function is a strictly increasing function of e over the interval (0, x(1) where x(1) = min(X1, X2, … Xn) and the likelihood function equals zero outside this interval. Therefore, the MLE exists and equals e = X(1)- (e) None of the above. The correct answer is (a) (b) (c) (d) (e) N/A
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
Related questions
Question
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MLE: Maximum likelihood estimation
![iid
(4) Let x1, X2, … , Xn
Pareto with the following density
2021x3, for x > 0,
felx) =
0,
otherwise,
where e e (0, ). Find an ML estimator of e if it exists.
The following "answers" have been proposed.
(a) The likelihood function of o is 2"o²n¡ [TE, X;. This is an even degree convex monomial defined over the whole real line.
1
Hence, it only has a minimum but no maximum. Hence, no MLE can exist.
(b) The likelihood function of e is 2"g2n/ [I1 X;. This is a polynomial that keeps increasing as e gets large over the positive
half of the real line. Hence, no MLE can exist.
(c) The likelihood function is a function of e over the interval (0, X(1)1 where x(1) = min(X1, X2, -. , Xn) and the likelihood
function equals zero outside this interval. Since the left end point of the interval is not included, the maximum may not
exist. Therefore, the MLE may not exist.
(d) The likelihood function is a strictly increasing function of o over the interval (0, X(1)] where x(1) = min(X1, X2, -. , Xn) and
the likelihood function equals zero outside this interval. Therefore, the MLE exists and equals
= X(1)-
(e) None of the above.
The correct answer is
(a)
(b)
(c)
(d)
(e)
N/A
(Select One)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2F59c32242-6449-4106-b259-950ce0f4f176%2Fix0vr73_processed.png&w=3840&q=75)
Transcribed Image Text:iid
(4) Let x1, X2, … , Xn
Pareto with the following density
2021x3, for x > 0,
felx) =
0,
otherwise,
where e e (0, ). Find an ML estimator of e if it exists.
The following "answers" have been proposed.
(a) The likelihood function of o is 2"o²n¡ [TE, X;. This is an even degree convex monomial defined over the whole real line.
1
Hence, it only has a minimum but no maximum. Hence, no MLE can exist.
(b) The likelihood function of e is 2"g2n/ [I1 X;. This is a polynomial that keeps increasing as e gets large over the positive
half of the real line. Hence, no MLE can exist.
(c) The likelihood function is a function of e over the interval (0, X(1)1 where x(1) = min(X1, X2, -. , Xn) and the likelihood
function equals zero outside this interval. Since the left end point of the interval is not included, the maximum may not
exist. Therefore, the MLE may not exist.
(d) The likelihood function is a strictly increasing function of o over the interval (0, X(1)] where x(1) = min(X1, X2, -. , Xn) and
the likelihood function equals zero outside this interval. Therefore, the MLE exists and equals
= X(1)-
(e) None of the above.
The correct answer is
(a)
(b)
(c)
(d)
(e)
N/A
(Select One)
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