Let X = (X1, X2, ..., Xn) be a sample of n observations each with a uniform in [0,0) density f (x, 0) = { ở if 0 < x < 0 0 else where 0 >0 is an unknown parameter. Denote the joint density by L(X, 0). a) Show that the family {L(X, 0)}, 0 > 0 has a monotone likelihood ratio in X(n). b) Show that the uniformly most powerful a-size test of Ho : 0 < 2 versus H1 : 0 > 2 is given by 1 if X(m) > 2(1 – a)- ={ o if X(m) < 2(1 – a)- %3D Justify all steps in your argument.
Let X = (X1, X2, ..., Xn) be a sample of n observations each with a uniform in [0,0) density f (x, 0) = { ở if 0 < x < 0 0 else where 0 >0 is an unknown parameter. Denote the joint density by L(X, 0). a) Show that the family {L(X, 0)}, 0 > 0 has a monotone likelihood ratio in X(n). b) Show that the uniformly most powerful a-size test of Ho : 0 < 2 versus H1 : 0 > 2 is given by 1 if X(m) > 2(1 – a)- ={ o if X(m) < 2(1 – a)- %3D Justify all steps in your argument.
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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Transcribed Image Text:Let X = (X1, X2, ..., X,) be a sample ofn observations each with a uniform in [0, 0) density
f (x, 0) = { i if 0 < x < 0
0 else
where 0 >0 is an unknown parameter. Denote the joint density by L(X, 0).
a) Show that the family {L(X,0)}, 0 >0 has a monotone likelihood ratio in X(n):
b) Show that the uniformly most powerful a-size test of Ho : 0 < 2 versus H : 0 > 2 is
given by
J 1 if X(m) > 2(1 - a) *
O if X(n) < 2(1 – a)
{
|
p* (X) =
Justify all steps in your argument.
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