application of Fisher-Neyman's factorization theorem
$begingroup$
Hi I am having some trouble understanding what kind of functions the right hand side of the factorization theorem is. My textbook states the theorem as:
$U$ is a sufficient statistic of a parameter $theta$ if
$L(theta) = g(u, theta) cdot h(y_{1}...y_{n})$
I have been looking at the example where $Y_{i}$ possesses the probability density function
$f(y_{i}|theta)= (1/theta)e^{-y_{i}/theta}$, 0 else
I can see how the likelihood function $L(theta)= frac{e^{-n bar{y}/theta}}{theta^{n}}$ is derived
But I cannot see why and how $g(bar{y},theta) = frac{e^{-n bar{y}/theta}}{theta^{n}}$
and how $h(y_{1}...y_{n})=1$
any explanation would be appreciated
statistics
asked Dec 8 '18 at 19:41
JensensJensens
245
$endgroup$
add a comment |
$begingroup$
Hi I am having some trouble understanding what kind of functions the right hand side of the factorization theorem is. My textbook states the theorem as:
$U$ is a sufficient statistic of a parameter $theta$ if
$L(theta) = g(u, theta) cdot h(y_{1}...y_{n})$
I have been looking at the example where $Y_{i}$ possesses the probability density function
$f(y_{i}|theta)= (1/theta)e^{-y_{i}/theta}$, 0 else
I can see how the likelihood function $L(theta)= frac{e^{-n bar{y}/theta}}{theta^{n}}$ is derived
But I cannot see why and how $g(bar{y},theta) = frac{e^{-n bar{y}/theta}}{theta^{n}}$
and how $h(y_{1}...y_{n})=1$
any explanation would be appreciated
statistics
asked Dec 8 '18 at 19:41
JensensJensens
245
$endgroup$
add a comment |
$begingroup$
Hi I am having some trouble understanding what kind of functions the right hand side of the factorization theorem is. My textbook states the theorem as:
$U$ is a sufficient statistic of a parameter $theta$ if
$L(theta) = g(u, theta) cdot h(y_{1}...y_{n})$
I have been looking at the example where $Y_{i}$ possesses the probability density function
$f(y_{i}|theta)= (1/theta)e^{-y_{i}/theta}$, 0 else
I can see how the likelihood function $L(theta)= frac{e^{-n bar{y}/theta}}{theta^{n}}$ is derived
But I cannot see why and how $g(bar{y},theta) = frac{e^{-n bar{y}/theta}}{theta^{n}}$
and how $h(y_{1}...y_{n})=1$
any explanation would be appreciated
statistics
asked Dec 8 '18 at 19:41
JensensJensens
245
$endgroup$
Hi I am having some trouble understanding what kind of functions the right hand side of the factorization theorem is. My textbook states the theorem as:
$U$ is a sufficient statistic of a parameter $theta$ if
$L(theta) = g(u, theta) cdot h(y_{1}...y_{n})$
I have been looking at the example where $Y_{i}$ possesses the probability density function
$f(y_{i}|theta)= (1/theta)e^{-y_{i}/theta}$, 0 else
I can see how the likelihood function $L(theta)= frac{e^{-n bar{y}/theta}}{theta^{n}}$ is derived
But I cannot see why and how $g(bar{y},theta) = frac{e^{-n bar{y}/theta}}{theta^{n}}$
and how $h(y_{1}...y_{n})=1$
any explanation would be appreciated
statistics
statistics
asked Dec 8 '18 at 19:41
JensensJensens
245
asked Dec 8 '18 at 19:41
JensensJensens
245
asked Dec 8 '18 at 19:41
JensensJensens
245
asked Dec 8 '18 at 19:41
JensensJensens
245
asked Dec 8 '18 at 19:41
JensensJensens
245
245
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%2f3031540%2fapplication-of-fisher-neymans-factorization-theorem%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%2f3031540%2fapplication-of-fisher-neymans-factorization-theorem%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
