Proving Pivots Statistics
$begingroup$
Suppose that $X_1, ldots, X_n$ are iid from
$$f_X(xmidθ) = atheta^{−a}x^{a−1}I(0 < x < theta)$$
, where $theta > 0$ is an unknown parameter and $a ge 1$ is a known constant.
Show that $T = X_{(n)}/theta$ is a pivotal quantity.
I know that $Q = Q(X, theta)$ is a pivot if the distribution of $Q$ does not depend on $θ$.
I honestly have no idea where to start with proving this. I have read the entire section in a textbook on pivots and even articles online. Any help is welcome.
statistics probability-distributions
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
$endgroup$
add a comment |
$begingroup$
Suppose that $X_1, ldots, X_n$ are iid from
$$f_X(xmidθ) = atheta^{−a}x^{a−1}I(0 < x < theta)$$
, where $theta > 0$ is an unknown parameter and $a ge 1$ is a known constant.
Show that $T = X_{(n)}/theta$ is a pivotal quantity.
I know that $Q = Q(X, theta)$ is a pivot if the distribution of $Q$ does not depend on $θ$.
I honestly have no idea where to start with proving this. I have read the entire section in a textbook on pivots and even articles online. Any help is welcome.
statistics probability-distributions
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
$endgroup$
1
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12
add a comment |
$begingroup$
Suppose that $X_1, ldots, X_n$ are iid from
$$f_X(xmidθ) = atheta^{−a}x^{a−1}I(0 < x < theta)$$
, where $theta > 0$ is an unknown parameter and $a ge 1$ is a known constant.
Show that $T = X_{(n)}/theta$ is a pivotal quantity.
I know that $Q = Q(X, theta)$ is a pivot if the distribution of $Q$ does not depend on $θ$.
I honestly have no idea where to start with proving this. I have read the entire section in a textbook on pivots and even articles online. Any help is welcome.
statistics probability-distributions
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
$endgroup$
Suppose that $X_1, ldots, X_n$ are iid from
$$f_X(xmidθ) = atheta^{−a}x^{a−1}I(0 < x < theta)$$
, where $theta > 0$ is an unknown parameter and $a ge 1$ is a known constant.
Show that $T = X_{(n)}/theta$ is a pivotal quantity.
I know that $Q = Q(X, theta)$ is a pivot if the distribution of $Q$ does not depend on $θ$.
I honestly have no idea where to start with proving this. I have read the entire section in a textbook on pivots and even articles online. Any help is welcome.
statistics probability-distributions
statistics probability-distributions
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
edited Dec 5 '18 at 19:04
StubbornAtom
6,06811239
6,06811239
asked Dec 5 '18 at 4:42
FireFire
6
asked Dec 5 '18 at 4:42
FireFire
6
asked Dec 5 '18 at 4:42
FireFire
6
6
1
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12
add a comment |
1
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12
1
1
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12
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%2f3026629%2fproving-pivots-statistics%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%2f3026629%2fproving-pivots-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
1
$begingroup$
Try to find distribution of $T$.
$endgroup$
– NCh
Dec 5 '18 at 7:12