How to show that $sum_{i=1}^m (X_i−X_m)^2$ and $sum_{i=1}^n(Y_i− Y_n)^2$ are independent
$begingroup$
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 '18 at 15:35
amWhy
1
asked Nov 27 '18 at 14:17
NewtNewt
207
$endgroup$
add a comment |
$begingroup$
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 '18 at 15:35
amWhy
1
asked Nov 27 '18 at 14:17
NewtNewt
207
$endgroup$
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39
add a comment |
$begingroup$
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
edited Nov 27 '18 at 15:35
amWhy
1
asked Nov 27 '18 at 14:17
NewtNewt
207
$endgroup$
Let $X_1,...,X_m$ be i.i.d. sample with $N(mu_1,sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(mu_2,2sigma^2)$.
Let $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$.
(b) Determine the values of $alpha$ and $beta$ for which $alpha S_x^2 + beta S_y^2$ will be an unbiased estimator with minimum variance.
It is not so difficult to solve if $S_x^2 = sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= sum_{i=1}^n(Y_i− Y_n)^2$ are independent. However I don't know how to show that they are independent... any possible helps??
independence sampling
independence sampling
edited Nov 27 '18 at 15:35
amWhy
1
asked Nov 27 '18 at 14:17
NewtNewt
207
edited Nov 27 '18 at 15:35
amWhy
1
asked Nov 27 '18 at 14:17
NewtNewt
207
edited Nov 27 '18 at 15:35
amWhy
1
edited Nov 27 '18 at 15:35
amWhy
1
edited Nov 27 '18 at 15:35
amWhy
1
1
asked Nov 27 '18 at 14:17
NewtNewt
207
asked Nov 27 '18 at 14:17
NewtNewt
207
asked Nov 27 '18 at 14:17
NewtNewt
207
207
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39
add a comment |
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39
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%2f3015818%2fhow-to-show-that-sum-i-1m-x-i%25e2%2588%2592x-m2-and-sum-i-1ny-i%25e2%2588%2592-y-n2-are-i%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%2f3015818%2fhow-to-show-that-sum-i-1m-x-i%25e2%2588%2592x-m2-and-sum-i-1ny-i%25e2%2588%2592-y-n2-are-i%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
$begingroup$
Are ${X_1,dots,X_m,Y_1,dots,Y_n}$ independent? If they are, then it's trivial.
$endgroup$
– Federico
Nov 27 '18 at 14:33
$begingroup$
Xi are independent and Yi are independent. But nothing's known about the independence between Xi and Yi
$endgroup$
– Newt
Nov 27 '18 at 14:37
$begingroup$
I think its pretty obvious from the statement that $X_i, Y_j$ are assumed to be independent.
$endgroup$
– leonbloy
Nov 27 '18 at 15:39