* (i) Let X,, .... X, be a random sample from the distribution with the PDF S(x: 0) = 2.r for 0
* (i) Let X,, .... X, be a random sample from the distribution with the PDF S(x: 0) = 2.r for 0
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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Question
![' (i) Let X,, .... X, be a random sample from
the distribution with the PDF
S(x: 0) =
2.r
for 0<x<0.
%3D
Determine the MLE M of 0 and show that (M, M/(1- Y)) is a 100y% CI for 0, where
0 <y<1.
(ii) Let X,, .... X, be an independent random sample from the uniform distribution on
[0, 0,] and let Y,, .... Y, be an independent random sample from the uniform distribution
on [0, 0,]. Derive the form of the likelihood ratio test of the hypothesis H,: 0, = 0, against
H: 0,#0, and express this test in terms of the statistic
%3D
mаx (Mx. Мy)
T =
min (Mx, My)
where Mx =max,<i<n X, and My=max1<ign Y.
By observing that under the hypothesis H, the distribution of T is independent of
0 = 0, = 0,, or otherwise, determine exactly the critical region for the test of size a.
%3D
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98b04692-e03c-419e-8fe1-534c478a92b9%2F59892651-f19c-4f58-be6b-f6bd2e6cb428%2F1mnwte_processed.jpeg&w=3840&q=75)
Transcribed Image Text:' (i) Let X,, .... X, be a random sample from
the distribution with the PDF
S(x: 0) =
2.r
for 0<x<0.
%3D
Determine the MLE M of 0 and show that (M, M/(1- Y)) is a 100y% CI for 0, where
0 <y<1.
(ii) Let X,, .... X, be an independent random sample from the uniform distribution on
[0, 0,] and let Y,, .... Y, be an independent random sample from the uniform distribution
on [0, 0,]. Derive the form of the likelihood ratio test of the hypothesis H,: 0, = 0, against
H: 0,#0, and express this test in terms of the statistic
%3D
mаx (Mx. Мy)
T =
min (Mx, My)
where Mx =max,<i<n X, and My=max1<ign Y.
By observing that under the hypothesis H, the distribution of T is independent of
0 = 0, = 0,, or otherwise, determine exactly the critical region for the test of size a.
%3D
%3D
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