Skip to main content

Questions tagged [cramer-rao]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
8 views

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 ...
Loco Citato's user avatar
1 vote
0 answers
48 views

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 ...
CfourPiO's user avatar
  • 235
0 votes
2 answers
53 views

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 ...
Luis Mendo's user avatar
  • 1,099
3 votes
1 answer
51 views

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....
Luis Mendo's user avatar
  • 1,099
1 vote
0 answers
42 views

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 ...
Lucas's user avatar
  • 11
0 votes
0 answers
43 views

Disproving the regularity condition of Cramer-Rao Lower bound

Let $X = (X_1,\cdots, X_n)$ where $X_1,\cdots,X_n$ be i.i.d from the uniform distribution $U(0,\theta)$ with $\theta>0$. I was asked to show the regularity condition of the Cramer-Rao lower bound: $...
Nothing's user avatar
  • 287
2 votes
1 answer
90 views

Does Cramer Rao bound depend on the sample size?

I have a confusion regarding sample size and CRB. CRB can be found for any estimator, even when it is biased. There are formulas to compute a "biased-CRB". There are also papers suggesting ...
CfourPiO's user avatar
  • 235
1 vote
1 answer
118 views

Cramér-Rao regular model

Why is the fact that the support of each $f$ belonging to a Cramér-Rao regular model does not depend on the parameter implied by the condition that the derivative with respect to the parameter of $f $ ...
Francesco Favuzza's user avatar
0 votes
0 answers
114 views

Cramer-Rao bound (CRB) and Root-Mean-Square-Error / Mean-Square-Error (RMSE / MSE)

My question is regarding the comparison between the CRB of a given vector parameter and RMSE/MSE obtained from Monte-Carlo (MC) simulation. The approach I used is this: For $\boldsymbol{\theta} \in \...
Zero's user avatar
  • 121
1 vote
0 answers
31 views

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 ...
DangerousTim's user avatar
2 votes
1 answer
236 views

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 ...
Hermi's user avatar
  • 747
0 votes
1 answer
250 views

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 $\...
Pramesh Pudasaini's user avatar
1 vote
1 answer
323 views

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 ...
Hermi's user avatar
  • 747
0 votes
1 answer
119 views

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. ...
Hermi's user avatar
  • 747
2 votes
0 answers
149 views

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 ...
Wil's user avatar
  • 21
2 votes
0 answers
106 views

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 ...
user avatar
8 votes
4 answers
1k views

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}{...
narip's user avatar
  • 185
2 votes
1 answer
314 views

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 $\...
narip's user avatar
  • 185
0 votes
0 answers
52 views

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 ...
Cooper's user avatar
  • 31
1 vote
1 answer
388 views

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_\...
The Pointer's user avatar
  • 2,086
5 votes
1 answer
319 views

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}) ...
The Pointer's user avatar
  • 2,086
0 votes
1 answer
534 views

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 $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...
The Pointer's user avatar
  • 2,086
3 votes
1 answer
2k views

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}...
Michael Baudin's user avatar
9 votes
3 answers
952 views

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.
a1a5a6's user avatar
  • 107
2 votes
0 answers
110 views

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 ...
dima_b's user avatar
  • 121