Concentration inequality applied for robust estimation of the mean
Problem: (Page 19 in "Vershynin, Roman (2018). High-Dimensional Probability. Cambridge University Press. ISBN 9781108415194.")
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / (delta varepsilon^2) )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability statistics median
asked Nov 15 '18 at 17:02
Rikeijin
949
add a comment |
Problem: (Page 19 in "Vershynin, Roman (2018). High-Dimensional Probability. Cambridge University Press. ISBN 9781108415194.")
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / (delta varepsilon^2) )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability statistics median
asked Nov 15 '18 at 17:02
Rikeijin
949
add a comment |
Problem: (Page 19 in "Vershynin, Roman (2018). High-Dimensional Probability. Cambridge University Press. ISBN 9781108415194.")
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / (delta varepsilon^2) )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability statistics median
asked Nov 15 '18 at 17:02
Rikeijin
949
Problem: (Page 19 in "Vershynin, Roman (2018). High-Dimensional Probability. Cambridge University Press. ISBN 9781108415194.")
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / (delta varepsilon^2) )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability statistics median
probability statistics median
asked Nov 15 '18 at 17:02
Rikeijin
949
asked Nov 15 '18 at 17:02
Rikeijin
949
edited 2 days ago
asked Nov 15 '18 at 17:02
Rikeijin
949
asked Nov 15 '18 at 17:02
Rikeijin
949
asked Nov 15 '18 at 17:02
Rikeijin
949
949
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%2f2999970%2fconcentration-inequality-applied-for-robust-estimation-of-the-mean%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f2999970%2fconcentration-inequality-applied-for-robust-estimation-of-the-mean%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
