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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?

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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 $\theta$ $\hat\theta(x_{1:n})$, for instance the MLE, has a limiting distribution as the sample size $n$ grows to infinity, for instance $\sqrt{n}(\hat\theta(X_{1:n})-\theta)$ 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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