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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Poisson: finding UMVUE for $\lambda + \lambda^2$ [closed]
Let $X_1,..., X_n$ be iid sample from the Poisson distribution with parameter $\lambda$. Find the UMVUE of $\lambda + \lambda^2$.
I know $T := \sum\limits_{i=1}^n X_i$ is complete and sufficient for $\...
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Find a function of $\theta$ so that there exists an unbiased estimator and the variance coincides with Cramér-Rao lower bound
Let $X_1,\dots, X_n$ be a random sample from the geometric distribution $P(X=x)=\theta(1-\theta)^x$ for $x=0,1,2,\dots$ where $0<\theta<1$.
Find a function of $\theta$, say $\tau=h(\theta)$ so ...
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Which Fisher information should I use for Cramer-Rao lower bound?
For $X_1,\dots, X_n$ iid sample from $X\sim Bernoulli(p)$. I try to verity that the estimator $\hat{p}=\bar{X}$ (sample mean) is the UMVUE for unknown parameter $p$.
I know that $\hat{p}$ is unbiased. ...
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Problem with the Fisher information matrix in case of N measurements of two observables
Let consider two observables, $x$ and $y$. Suppose that $y$ depends on the independent variable $x$ through the model $m(x; \boldsymbol{\theta})$, where $\boldsymbol{\theta}$ is a vector of model ...
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Do unbiased L2 consistent, consistent in probability, or almost sure consistent estimator have a asymptotic variance equal to rao-cramer or better?
Does a L2 consistent, consistent in probability, or almost sure consistent estimator have a asymptotic variance equal to rao-cramer or better?
With almost-sure consistency, why doesn't the estimator ...
user318514
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Why the variance of Maximum Likelihood Estimator(MLE) will be less than Cramer-Rao Lower Bound(CRLB)?
Consider this example. Suppose we have three events to happen with probability $p_1=p_2=\frac{1}{2}\sin ^2\theta ,p_3=\cos ^2\theta $ respectively. And we suppose the true value $\theta _0=\frac{\pi}{...
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Attainablility of Cramer Rao Bound with function of multi-parameters?
Suppose we have multivariables ${\boldsymbol {\theta }}=\left[\theta _{1},\theta _{2},\dots ,\theta _{d}\right]^{T}\in {\mathbb {R}}^{d}$, and we want to estimate the function of parameters $\...
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Rao Cramèr Lower Bound problem
Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
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Why does $T$ being an unbiased estimator for $g(\theta)$ imply that $g(\theta) = ET = \int T(\mathbf{y}) f_\theta(\mathbf{y}) \ d\mathbf{y}$?
I am currently studying the Cramer-Rao lower bound. My notes say the following:
Theorem: Cramer-Rao lower bound
Let $Y_1, \dots, Y_n$ have a joint distribution $f_\theta (\mathbf{y})$, where $f_\...
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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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Worst-case error related to Cramer-Rao bound
Asked this previously on Math.SE, maybe this fits here.
I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of ...
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