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Confused about Cramer-Rao lower bound and CLT

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3

1

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Learning statistics for application in the physical sciences. I am confused about the cramer-rao (CR) bound vs central limit theorem for estimating the variance of the sample mean. I thought that once we have enough samples, we can use CLT regardless of the shape of the distribution. When would we use the CR bound to calculate the confidence intervals vs CLT? Is it only when the number of samples are low which would lead the CLT to not converge?

inference central-limit-theorem

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asked Mar 21 at 15:01

ChuckChuck

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3

1

$begingroup$

Learning statistics for application in the physical sciences. I am confused about the cramer-rao (CR) bound vs central limit theorem for estimating the variance of the sample mean. I thought that once we have enough samples, we can use CLT regardless of the shape of the distribution. When would we use the CR bound to calculate the confidence intervals vs CLT? Is it only when the number of samples are low which would lead the CLT to not converge?

inference central-limit-theorem

share|cite|improve this question

asked Mar 21 at 15:01

ChuckChuck

161

New contributor

Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Learning statistics for application in the physical sciences. I am confused about the cramer-rao (CR) bound vs central limit theorem for estimating the variance of the sample mean. I thought that once we have enough samples, we can use CLT regardless of the shape of the distribution. When would we use the CR bound to calculate the confidence intervals vs CLT? Is it only when the number of samples are low which would lead the CLT to not converge?

inference central-limit-theorem

share|cite|improve this question

asked Mar 21 at 15:01

ChuckChuck

161

New contributor

Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked Mar 21 at 15:01

ChuckChuck

161

New contributor

Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 21 at 15:01

ChuckChuck

161

New contributor

Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 21 at 15:01

ChuckChuck

161

New contributor

Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 21 at 15:01

ChuckChuck

161

1 Answer
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The Cramèr-Rao lower bound (or Fréchet-Darmois-Cramèr-Rao lower bound) is a lower bound on the variance of a collection of estimators, for instance unbiased estimators or estimators with a given bias. In some cases, there exists an estimator within the collection that meets this lower bound, but in other cases, the lower bound may be strict. Using a Cramèr-Rao lower bound to construct a confidence interval is a very crude approach to confidence intervals.

The CLT is a version of a limiting theorem, which means that an estimator of $theta$ $hattheta(x_1:n)$, for instance the MLE, has a limiting distribution as the sample size $n$ grows to infinity, for instance $sqrtn(hattheta(X_1:n)-theta)$ converges in distribution to a $mathcal N(0,sigma^2(theta))$ distribution. For a collection of estimators of $theta$, asymptotic variances $sigma^2(theta)$ may be compared, but this does not imply anything at the finite $n$ level.

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edited Mar 21 at 19:33

answered Mar 21 at 16:55

Xi'anXi'an

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7

$begingroup$

The Cramèr-Rao lower bound (or Fréchet-Darmois-Cramèr-Rao lower bound) is a lower bound on the variance of a collection of estimators, for instance unbiased estimators or estimators with a given bias. In some cases, there exists an estimator within the collection that meets this lower bound, but in other cases, the lower bound may be strict. Using a Cramèr-Rao lower bound to construct a confidence interval is a very crude approach to confidence intervals.

The CLT is a version of a limiting theorem, which means that an estimator of $theta$ $hattheta(x_1:n)$, for instance the MLE, has a limiting distribution as the sample size $n$ grows to infinity, for instance $sqrtn(hattheta(X_1:n)-theta)$ converges in distribution to a $mathcal N(0,sigma^2(theta))$ distribution. For a collection of estimators of $theta$, asymptotic variances $sigma^2(theta)$ may be compared, but this does not imply anything at the finite $n$ level.

share|cite|improve this answer

edited Mar 21 at 19:33

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364

$endgroup$

add a comment | 

7

$begingroup$

The Cramèr-Rao lower bound (or Fréchet-Darmois-Cramèr-Rao lower bound) is a lower bound on the variance of a collection of estimators, for instance unbiased estimators or estimators with a given bias. In some cases, there exists an estimator within the collection that meets this lower bound, but in other cases, the lower bound may be strict. Using a Cramèr-Rao lower bound to construct a confidence interval is a very crude approach to confidence intervals.

The CLT is a version of a limiting theorem, which means that an estimator of $theta$ $hattheta(x_1:n)$, for instance the MLE, has a limiting distribution as the sample size $n$ grows to infinity, for instance $sqrtn(hattheta(X_1:n)-theta)$ converges in distribution to a $mathcal N(0,sigma^2(theta))$ distribution. For a collection of estimators of $theta$, asymptotic variances $sigma^2(theta)$ may be compared, but this does not imply anything at the finite $n$ level.

share|cite|improve this answer

edited Mar 21 at 19:33

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364

$endgroup$

add a comment | 

$begingroup$

The Cramèr-Rao lower bound (or Fréchet-Darmois-Cramèr-Rao lower bound) is a lower bound on the variance of a collection of estimators, for instance unbiased estimators or estimators with a given bias. In some cases, there exists an estimator within the collection that meets this lower bound, but in other cases, the lower bound may be strict. Using a Cramèr-Rao lower bound to construct a confidence interval is a very crude approach to confidence intervals.

The CLT is a version of a limiting theorem, which means that an estimator of $theta$ $hattheta(x_1:n)$, for instance the MLE, has a limiting distribution as the sample size $n$ grows to infinity, for instance $sqrtn(hattheta(X_1:n)-theta)$ converges in distribution to a $mathcal N(0,sigma^2(theta))$ distribution. For a collection of estimators of $theta$, asymptotic variances $sigma^2(theta)$ may be compared, but this does not imply anything at the finite $n$ level.

share|cite|improve this answer

edited Mar 21 at 19:33

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364

$endgroup$

share|cite|improve this answer

edited Mar 21 at 19:33

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364

answered Mar 21 at 16:55

Xi'anXi'an

58.9k897364