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Confused about Cramer-Rao lower bound and CLT
How to demonstrate failure of CLT in R?Confidence intervals based on the CLT: ever useful?Upper bound for sampling fraction for CLT to holdCramer-Rao Lower BoundConfidence intervals and Central Limit Theorem with only one sampleOnline course for inferential statisticsHow does the Central Limit Theorem show that the Binomial Distribution is approximately Normal for a large value of n?CLT and Determining Normality Without DatasetDifference between the Law of Large Numbers and the Central Limit Theorem in layman's term?Confused about Assumptions of Central Limit Theorem
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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
asked 11 hours ago
ChuckChuck
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Chuck is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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
asked 11 hours ago
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
asked 11 hours ago
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.
$endgroup$
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
inference central-limit-theorem
asked 11 hours ago
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 11 hours ago
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 11 hours ago
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 11 hours ago
ChuckChuck
161
asked 11 hours ago
ChuckChuck
161
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.
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.
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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.
answered 9 hours ago
Xi'anXi'an
58.6k897363
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1 Answer
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1 Answer
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$begingroup$
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.
answered 9 hours ago
Xi'anXi'an
58.6k897363
$endgroup$
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.
answered 9 hours ago
Xi'anXi'an
58.6k897363
$endgroup$
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.
answered 9 hours ago
Xi'anXi'an
58.6k897363
$endgroup$
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.
answered 9 hours ago
Xi'anXi'an
58.6k897363
edited 7 hours ago
answered 9 hours ago
Xi'anXi'an
58.6k897363
answered 9 hours ago
Xi'anXi'an
58.6k897363
answered 9 hours ago
Xi'anXi'an
58.6k897363
58.6k897363
add a comment |
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