Mixing
Likelihood ratio test at level $alpha.$ (Verification)
$begingroup$
We have an observation $X_1$ distributed with a density function
$$f_{theta}(x)=frac{theta}{(x+theta)^2}$$
for $xgeq 0$, otherwise $f_{theta}(x)=0.$ Here we assume that $theta>0.$ We want to test the Hypothesis $H_0:theta = theta_0$ vs. $H_1:theta = theta_1$ where $theta_0<theta_1.$
For this, we write the likelihood ratio:
$$Lambda(X_1,H_1,H_0)=frac{mathcal{L}(theta_1;X_1)}{mathcal{L}(theta_0;X_1)} = frac{theta_1}{theta_0}cdot left(frac{x_1+theta_0}{x_1+theta_1}right)^2.$$
Then the derivative of this ratio is
$$frac{2left(x_1+theta_0right)left(theta_1-theta_0right)}{left(x_1+theta_1right)^3}>0$$
and so the ratio is increasing as a function of $x_1.$
This means that the likelihood ratio test function is
$$ phi_{alpha}(x) =
begin{cases}
1 & x_1>c_{alpha} \
gamma_{alpha} & x_1 =c_{alpha} \
0 & x_1 <c_{alpha}
end{cases}
$$
where we need to find $c_{alpha}$ and $gamma_{alpha}.$ We know that if the significance of the test is $alpha$ then
$$alpha = E_{theta_{0}}[phi_{alpha}(X)] = P_{theta_0}(x_1<c_{alpha}) + gamma_{alpha}P_{theta_0}(x_1=c_{alpha}).$$
We know that $P_{theta_0}(x_1<c_{alpha}) = int_{0}^{c_{alpha}}frac{theta_0}{(theta_0+x)^2}dx = frac{c_{alpha}}{c_{alpha} + theta_0}.$
We thus see that taking $gamma_{alpha}=0$ and $c_{alpha} =frac{alpha theta_0}{1-alpha}$ works.
This gives us our most powerful test at level $alpha$
$$ phi_{alpha}(x) =
begin{cases}
1 & x>frac{alpha theta_0}{1-alpha} \
0 & text{otherwise}
end{cases}.
$$
I finally compute the Type II risk which is
$$P_{theta_1}(phi(x) = 0)= P_{theta_1}left(xleq frac{alpha theta_0}{1-alpha}right) = frac{alpha theta_0}{alpha theta_0 + (1-alpha)theta_1}.$$
Are these calculations correct?
statistics maximum-likelihood
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
$endgroup$
add a comment |
$begingroup$
We have an observation $X_1$ distributed with a density function
$$f_{theta}(x)=frac{theta}{(x+theta)^2}$$
for $xgeq 0$, otherwise $f_{theta}(x)=0.$ Here we assume that $theta>0.$ We want to test the Hypothesis $H_0:theta = theta_0$ vs. $H_1:theta = theta_1$ where $theta_0<theta_1.$
For this, we write the likelihood ratio:
$$Lambda(X_1,H_1,H_0)=frac{mathcal{L}(theta_1;X_1)}{mathcal{L}(theta_0;X_1)} = frac{theta_1}{theta_0}cdot left(frac{x_1+theta_0}{x_1+theta_1}right)^2.$$
Then the derivative of this ratio is
$$frac{2left(x_1+theta_0right)left(theta_1-theta_0right)}{left(x_1+theta_1right)^3}>0$$
and so the ratio is increasing as a function of $x_1.$
This means that the likelihood ratio test function is
$$ phi_{alpha}(x) =
begin{cases}
1 & x_1>c_{alpha} \
gamma_{alpha} & x_1 =c_{alpha} \
0 & x_1 <c_{alpha}
end{cases}
$$
where we need to find $c_{alpha}$ and $gamma_{alpha}.$ We know that if the significance of the test is $alpha$ then
$$alpha = E_{theta_{0}}[phi_{alpha}(X)] = P_{theta_0}(x_1<c_{alpha}) + gamma_{alpha}P_{theta_0}(x_1=c_{alpha}).$$
We know that $P_{theta_0}(x_1<c_{alpha}) = int_{0}^{c_{alpha}}frac{theta_0}{(theta_0+x)^2}dx = frac{c_{alpha}}{c_{alpha} + theta_0}.$
We thus see that taking $gamma_{alpha}=0$ and $c_{alpha} =frac{alpha theta_0}{1-alpha}$ works.
This gives us our most powerful test at level $alpha$
$$ phi_{alpha}(x) =
begin{cases}
1 & x>frac{alpha theta_0}{1-alpha} \
0 & text{otherwise}
end{cases}.
$$
I finally compute the Type II risk which is
$$P_{theta_1}(phi(x) = 0)= P_{theta_1}left(xleq frac{alpha theta_0}{1-alpha}right) = frac{alpha theta_0}{alpha theta_0 + (1-alpha)theta_1}.$$
Are these calculations correct?
statistics maximum-likelihood
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
$endgroup$
add a comment |
$begingroup$
We have an observation $X_1$ distributed with a density function
$$f_{theta}(x)=frac{theta}{(x+theta)^2}$$
for $xgeq 0$, otherwise $f_{theta}(x)=0.$ Here we assume that $theta>0.$ We want to test the Hypothesis $H_0:theta = theta_0$ vs. $H_1:theta = theta_1$ where $theta_0<theta_1.$
For this, we write the likelihood ratio:
$$Lambda(X_1,H_1,H_0)=frac{mathcal{L}(theta_1;X_1)}{mathcal{L}(theta_0;X_1)} = frac{theta_1}{theta_0}cdot left(frac{x_1+theta_0}{x_1+theta_1}right)^2.$$
Then the derivative of this ratio is
$$frac{2left(x_1+theta_0right)left(theta_1-theta_0right)}{left(x_1+theta_1right)^3}>0$$
and so the ratio is increasing as a function of $x_1.$
This means that the likelihood ratio test function is
$$ phi_{alpha}(x) =
begin{cases}
1 & x_1>c_{alpha} \
gamma_{alpha} & x_1 =c_{alpha} \
0 & x_1 <c_{alpha}
end{cases}
$$
where we need to find $c_{alpha}$ and $gamma_{alpha}.$ We know that if the significance of the test is $alpha$ then
$$alpha = E_{theta_{0}}[phi_{alpha}(X)] = P_{theta_0}(x_1<c_{alpha}) + gamma_{alpha}P_{theta_0}(x_1=c_{alpha}).$$
We know that $P_{theta_0}(x_1<c_{alpha}) = int_{0}^{c_{alpha}}frac{theta_0}{(theta_0+x)^2}dx = frac{c_{alpha}}{c_{alpha} + theta_0}.$
We thus see that taking $gamma_{alpha}=0$ and $c_{alpha} =frac{alpha theta_0}{1-alpha}$ works.
This gives us our most powerful test at level $alpha$
$$ phi_{alpha}(x) =
begin{cases}
1 & x>frac{alpha theta_0}{1-alpha} \
0 & text{otherwise}
end{cases}.
$$
I finally compute the Type II risk which is
$$P_{theta_1}(phi(x) = 0)= P_{theta_1}left(xleq frac{alpha theta_0}{1-alpha}right) = frac{alpha theta_0}{alpha theta_0 + (1-alpha)theta_1}.$$
Are these calculations correct?
statistics maximum-likelihood
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
$endgroup$
We have an observation $X_1$ distributed with a density function
$$f_{theta}(x)=frac{theta}{(x+theta)^2}$$
for $xgeq 0$, otherwise $f_{theta}(x)=0.$ Here we assume that $theta>0.$ We want to test the Hypothesis $H_0:theta = theta_0$ vs. $H_1:theta = theta_1$ where $theta_0<theta_1.$
For this, we write the likelihood ratio:
$$Lambda(X_1,H_1,H_0)=frac{mathcal{L}(theta_1;X_1)}{mathcal{L}(theta_0;X_1)} = frac{theta_1}{theta_0}cdot left(frac{x_1+theta_0}{x_1+theta_1}right)^2.$$
Then the derivative of this ratio is
$$frac{2left(x_1+theta_0right)left(theta_1-theta_0right)}{left(x_1+theta_1right)^3}>0$$
and so the ratio is increasing as a function of $x_1.$
This means that the likelihood ratio test function is
$$ phi_{alpha}(x) =
begin{cases}
1 & x_1>c_{alpha} \
gamma_{alpha} & x_1 =c_{alpha} \
0 & x_1 <c_{alpha}
end{cases}
$$
where we need to find $c_{alpha}$ and $gamma_{alpha}.$ We know that if the significance of the test is $alpha$ then
$$alpha = E_{theta_{0}}[phi_{alpha}(X)] = P_{theta_0}(x_1<c_{alpha}) + gamma_{alpha}P_{theta_0}(x_1=c_{alpha}).$$
We know that $P_{theta_0}(x_1<c_{alpha}) = int_{0}^{c_{alpha}}frac{theta_0}{(theta_0+x)^2}dx = frac{c_{alpha}}{c_{alpha} + theta_0}.$
We thus see that taking $gamma_{alpha}=0$ and $c_{alpha} =frac{alpha theta_0}{1-alpha}$ works.
This gives us our most powerful test at level $alpha$
$$ phi_{alpha}(x) =
begin{cases}
1 & x>frac{alpha theta_0}{1-alpha} \
0 & text{otherwise}
end{cases}.
$$
I finally compute the Type II risk which is
$$P_{theta_1}(phi(x) = 0)= P_{theta_1}left(xleq frac{alpha theta_0}{1-alpha}right) = frac{alpha theta_0}{alpha theta_0 + (1-alpha)theta_1}.$$
Are these calculations correct?
statistics maximum-likelihood
statistics maximum-likelihood
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
asked Jan 21 at 12:30
Hello_WorldHello_World
4,14621931
4,14621931
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3081820%2flikelihood-ratio-test-at-level-alpha-verification%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3081820%2flikelihood-ratio-test-at-level-alpha-verification%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
