Density of a diffeomorphic image of a random vector
$begingroup$
Let $X = (X_1, dots, X_n)$ be a random vector and $fcolon mathbb{R}^nrightarrow mathbb{R}^n$ an a.e.-differentiable mapping.
- If $f$ is a diffeomorphism, is it possible to derive the density of $f(X)$?
- When does $f(X)$ have no density w.r.t. the $n$-dimensional Lebesgue measure?
In 1., it looks like integration by substitution should do the trick. Still, integration by substitution will give us integration over the wrong set, i.e. $f^{-1}(A)$ instead of $A$.
In 2., a sufficient condition is that at every point the derivative is degenerate, then by Sard's theorem the image of $f$ has measure zero, so it clearly has no density.
probability probability-distributions density-function
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
$endgroup$
add a comment |
$begingroup$
Let $X = (X_1, dots, X_n)$ be a random vector and $fcolon mathbb{R}^nrightarrow mathbb{R}^n$ an a.e.-differentiable mapping.
- If $f$ is a diffeomorphism, is it possible to derive the density of $f(X)$?
- When does $f(X)$ have no density w.r.t. the $n$-dimensional Lebesgue measure?
In 1., it looks like integration by substitution should do the trick. Still, integration by substitution will give us integration over the wrong set, i.e. $f^{-1}(A)$ instead of $A$.
In 2., a sufficient condition is that at every point the derivative is degenerate, then by Sard's theorem the image of $f$ has measure zero, so it clearly has no density.
probability probability-distributions density-function
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
$endgroup$
add a comment |
$begingroup$
Let $X = (X_1, dots, X_n)$ be a random vector and $fcolon mathbb{R}^nrightarrow mathbb{R}^n$ an a.e.-differentiable mapping.
- If $f$ is a diffeomorphism, is it possible to derive the density of $f(X)$?
- When does $f(X)$ have no density w.r.t. the $n$-dimensional Lebesgue measure?
In 1., it looks like integration by substitution should do the trick. Still, integration by substitution will give us integration over the wrong set, i.e. $f^{-1}(A)$ instead of $A$.
In 2., a sufficient condition is that at every point the derivative is degenerate, then by Sard's theorem the image of $f$ has measure zero, so it clearly has no density.
probability probability-distributions density-function
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
$endgroup$
Let $X = (X_1, dots, X_n)$ be a random vector and $fcolon mathbb{R}^nrightarrow mathbb{R}^n$ an a.e.-differentiable mapping.
- If $f$ is a diffeomorphism, is it possible to derive the density of $f(X)$?
- When does $f(X)$ have no density w.r.t. the $n$-dimensional Lebesgue measure?
In 1., it looks like integration by substitution should do the trick. Still, integration by substitution will give us integration over the wrong set, i.e. $f^{-1}(A)$ instead of $A$.
In 2., a sufficient condition is that at every point the derivative is degenerate, then by Sard's theorem the image of $f$ has measure zero, so it clearly has no density.
probability probability-distributions density-function
probability probability-distributions density-function
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
asked Jan 10 at 19:52
Abraham SlavitskyAbraham Slavitsky
12
12
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%2f3069100%2fdensity-of-a-diffeomorphic-image-of-a-random-vector%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%2f3069100%2fdensity-of-a-diffeomorphic-image-of-a-random-vector%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
