How to work out the variance of this estimator?
$begingroup$
We want the variance of this biased estimator.
The answer is: $$σ^2/4n$$
Why is it not this? $$σ^2/2n$$
I.e. wouldn't it be $$1/2n^2 * n * σ^2$$
Any clarification appreciated.
Thanks!
statistics variance parameter-estimation
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
$endgroup$
add a comment |
$begingroup$
We want the variance of this biased estimator.
The answer is: $$σ^2/4n$$
Why is it not this? $$σ^2/2n$$
I.e. wouldn't it be $$1/2n^2 * n * σ^2$$
Any clarification appreciated.
Thanks!
statistics variance parameter-estimation
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
$endgroup$
add a comment |
$begingroup$
We want the variance of this biased estimator.
The answer is: $$σ^2/4n$$
Why is it not this? $$σ^2/2n$$
I.e. wouldn't it be $$1/2n^2 * n * σ^2$$
Any clarification appreciated.
Thanks!
statistics variance parameter-estimation
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
$endgroup$
We want the variance of this biased estimator.
The answer is: $$σ^2/4n$$
Why is it not this? $$σ^2/2n$$
I.e. wouldn't it be $$1/2n^2 * n * σ^2$$
Any clarification appreciated.
Thanks!
statistics variance parameter-estimation
statistics variance parameter-estimation
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
asked May 20 '18 at 17:09
MathsHelpMathsHelp
16810
16810
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Note that, if $lambdain Bbb R$, then $text{Var}(lambda X) = lambda^{bf{2}}text{Var}(X)$, not $lambdatext{Var}(X)$.
Thus
$$text{Var}(hat{theta}) = left(frac{1}{2n}right)^2.text{Var}left(sum_{i=1}^n X_iright) = frac{1}{4n^2}.n.sigma^2 = frac{sigma^2}{4n}.$$
answered May 20 '18 at 17:22
pafpaf
4,0281824
$endgroup$
add a comment |
Your Answer
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%2f2789000%2fhow-to-work-out-the-variance-of-this-estimator%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
$begingroup$
Note that, if $lambdain Bbb R$, then $text{Var}(lambda X) = lambda^{bf{2}}text{Var}(X)$, not $lambdatext{Var}(X)$.
Thus
$$text{Var}(hat{theta}) = left(frac{1}{2n}right)^2.text{Var}left(sum_{i=1}^n X_iright) = frac{1}{4n^2}.n.sigma^2 = frac{sigma^2}{4n}.$$
answered May 20 '18 at 17:22
pafpaf
4,0281824
$endgroup$
add a comment |
$begingroup$
Note that, if $lambdain Bbb R$, then $text{Var}(lambda X) = lambda^{bf{2}}text{Var}(X)$, not $lambdatext{Var}(X)$.
Thus
$$text{Var}(hat{theta}) = left(frac{1}{2n}right)^2.text{Var}left(sum_{i=1}^n X_iright) = frac{1}{4n^2}.n.sigma^2 = frac{sigma^2}{4n}.$$
answered May 20 '18 at 17:22
pafpaf
4,0281824
$endgroup$
add a comment |
$begingroup$
Note that, if $lambdain Bbb R$, then $text{Var}(lambda X) = lambda^{bf{2}}text{Var}(X)$, not $lambdatext{Var}(X)$.
Thus
$$text{Var}(hat{theta}) = left(frac{1}{2n}right)^2.text{Var}left(sum_{i=1}^n X_iright) = frac{1}{4n^2}.n.sigma^2 = frac{sigma^2}{4n}.$$
answered May 20 '18 at 17:22
pafpaf
4,0281824
$endgroup$
Note that, if $lambdain Bbb R$, then $text{Var}(lambda X) = lambda^{bf{2}}text{Var}(X)$, not $lambdatext{Var}(X)$.
Thus
$$text{Var}(hat{theta}) = left(frac{1}{2n}right)^2.text{Var}left(sum_{i=1}^n X_iright) = frac{1}{4n^2}.n.sigma^2 = frac{sigma^2}{4n}.$$
answered May 20 '18 at 17:22
pafpaf
4,0281824
edited Jan 9 at 6:01
answered May 20 '18 at 17:22
pafpaf
4,0281824
answered May 20 '18 at 17:22
pafpaf
4,0281824
answered May 20 '18 at 17:22
pafpaf
4,0281824
4,0281824
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%2f2789000%2fhow-to-work-out-the-variance-of-this-estimator%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
