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# Questions tagged [cramer-rao]

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### Equivalence between CRLB and uncertainty propagation formula

I already asked this question in the "Mathematics" stackexchange, but apparently did not find the right audience, so I am duplicating my question here, hoping someone might be of help. My ...
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### How to find the Cramer-Rao bound for a biased estimator when the bias is not known in closed form

I have a maximum likelihood problem where I estimate two parameters. The likelihood function is that of an exponential distribution with mean $\lambda$. The parameters are parameters of this $\lambda$....
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### For some $\tau=\tau(\theta)$, there exists an unbiased estimator (UMVUE), then the distribution belongs to an exponential family

I read the textbook in Cramer-Rao lower bound (CRLB). Here is a theroem For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the ...
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### How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?

Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
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### How can one show that $\bar{X}$ is the best unbiased estimator for $\lambda$ without using the Cramèr-Rao lower bound?

Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] = \sigma^2$, where $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...
I search a detailed proof of the multivariate Cramer-Rao inequality in the general case where the estimator is not necessarily unbiased. Let $T(X)$ be an estimator of the parameter \$\theta\in\mathbb{R}...