# Questions tagged [cramer-rao]

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### How to deal with Bias Gradient Matrix for biased CRB(Cramér–Rao bound) calculation if the gradient matrix is m-by-n but $m \neq n$?

I am doing a model for collabrative localization and using the CRB(Cramér–Rao bound) as the localization performance measurement. I want to consider interference caused by NLOS and clutter, therefore ...
1 vote
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### Standard practice to show Biased CRBs

I have a problem with four-parameter estimation. I have derived the variances for the estimated parameters using Monte Carlo simulations (numerical ones) and theoretical ones using the inverse of the ...
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### Cramér-Rao bound when the samples come from two distributions

Is there a version of the Cramér-Rao bound when samples are independent but not identically distributed? More specifically, I am considering a sample set that is divided in two subsets, each subset ...
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### Cramér-Rao / Wolfowitz bound with nuisance parameter

Let $F$ be a distribution with two parameters, $\theta$ and $\phi$, whose values are non-random but unknown. Consider a sampling procedure in which $N$ samples $x_1, \ldots x_N$ are obtained from i.i....
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1 vote
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### Derive Cramer-Rao lower bound for $Var(\hat{\theta})$ given that $\mathbb{E}[\hat{\theta}U]=1$

I am trying to derive the Cramer-Rao lower bound for $Var(\hat{\theta})$ given that we already know $\mathbb{E}[U]=0$, $Var(U)=I(\theta)$ and $\mathbb{E}[\hat{\theta}U]=1$. I am struggling with using ...
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### Which Fisher information to use to obtain Cramer-Rao bound in expectation-maximization?

I have a rather limited understanding of statistical estimation theory so I apologize if my question is strange or trivial. Say I have an expectation-maximization-based algorithm for determining the ...
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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 $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...
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### Proof of the multivariate Cramer-Rao inequality

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}...
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### Why is the Cramer-Rao Lower Bound (CRLB) inverse of the Fisher Information I(θ)?

Why is the Cramer-Rao Lower Bound (CRLB) inverse of the Fisher Information I(θ)? Could someone provide an intuitive explanation? I am having trouble understanding the concept.
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