Complete statistics
$begingroup$
Let $(x_{1}, ldots, x_{n})$ be a sample from some known distribution $F_{theta}$ with unknown parameter $theta$ and let $$mathbb{E}_{theta}{X_{i}} = theta$$
Let $T$ be an identity function of a sample e.g. for the given sample $T$ returns the sample itself. How to prove that $T$ is not a complete statistics?
By definition, the complete statistics is a function $T$ such that for any measurable $g$ such that $$mathbb{E}(g(T)) = 0$$
for any $theta$
$$mathbb{P}_{theta}(g(T) = 0) = 1$$
for any $theta$.
I guess that the problem is quite easy but i cannot find an easy way to attack it. Are there any hints that might help?
statistics statistical-inference
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
$endgroup$
add a comment |
$begingroup$
Let $(x_{1}, ldots, x_{n})$ be a sample from some known distribution $F_{theta}$ with unknown parameter $theta$ and let $$mathbb{E}_{theta}{X_{i}} = theta$$
Let $T$ be an identity function of a sample e.g. for the given sample $T$ returns the sample itself. How to prove that $T$ is not a complete statistics?
By definition, the complete statistics is a function $T$ such that for any measurable $g$ such that $$mathbb{E}(g(T)) = 0$$
for any $theta$
$$mathbb{P}_{theta}(g(T) = 0) = 1$$
for any $theta$.
I guess that the problem is quite easy but i cannot find an easy way to attack it. Are there any hints that might help?
statistics statistical-inference
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
$endgroup$
add a comment |
$begingroup$
Let $(x_{1}, ldots, x_{n})$ be a sample from some known distribution $F_{theta}$ with unknown parameter $theta$ and let $$mathbb{E}_{theta}{X_{i}} = theta$$
Let $T$ be an identity function of a sample e.g. for the given sample $T$ returns the sample itself. How to prove that $T$ is not a complete statistics?
By definition, the complete statistics is a function $T$ such that for any measurable $g$ such that $$mathbb{E}(g(T)) = 0$$
for any $theta$
$$mathbb{P}_{theta}(g(T) = 0) = 1$$
for any $theta$.
I guess that the problem is quite easy but i cannot find an easy way to attack it. Are there any hints that might help?
statistics statistical-inference
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
$endgroup$
Let $(x_{1}, ldots, x_{n})$ be a sample from some known distribution $F_{theta}$ with unknown parameter $theta$ and let $$mathbb{E}_{theta}{X_{i}} = theta$$
Let $T$ be an identity function of a sample e.g. for the given sample $T$ returns the sample itself. How to prove that $T$ is not a complete statistics?
By definition, the complete statistics is a function $T$ such that for any measurable $g$ such that $$mathbb{E}(g(T)) = 0$$
for any $theta$
$$mathbb{P}_{theta}(g(T) = 0) = 1$$
for any $theta$.
I guess that the problem is quite easy but i cannot find an easy way to attack it. Are there any hints that might help?
statistics statistical-inference
statistics statistical-inference
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
asked Dec 17 '18 at 18:51
hyperkahlerhyperkahler
1,471714
1,471714
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
$endgroup$
add a comment |
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%2f3044299%2fcomplete-statistics%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
$endgroup$
add a comment |
$begingroup$
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
$endgroup$
add a comment |
$begingroup$
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
$endgroup$
Take $g(T)=X_1-X_2$. Then $E(g(T))=0$ but $g$ is not identically $0$ with probability $1$.
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
answered Dec 17 '18 at 18:58
pulpfictionalpulpfictional
835
835
add a comment |
add a comment |
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%2f3044299%2fcomplete-statistics%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
