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
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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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