# Questions tagged [fisher-information]

The Fisher information measures the curvature of the log-likelihood and can be used to assess the efficiency of estimators.

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### Cramér-Rao Lower Bound & Fisher information - error in textbook?

I'm currently reading the textbook "Statistics for Mathematician" from Victor Panaretos. On page 65, the author presents the following equation for the Cramér-Rao Lower Bound (Note: I set the ...
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### How to compute variance/confidence intervals from Fisher information matrix. Mistake in this document?

https://www.stat.umn.edu/geyer/5931/mle/mle.pdf In this document (by the great Geyer nonetheless!) it calculates confidence intervals using the Fisher matrix: But the standard deviation is not the ...
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### Singular Fisher Information and nuisance parameters

I am interested in a problem where the Fisher Information matrix is singular and I want to treat some of the parameters as nuisance parameters. If the FI matrix is singular, that means that some ...
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### Confidence interval for the 95th percentile of the normal distribution

Let $X_1, .., X_n \sim Normal(\mu, \sigma^2)$. Let $\tau$ be the 95th percentile of this distribution. Thus, $P(X_i < \tau) = 0.95$. What is the $1 - \alpha$ confidence interval for $\tau$? I ...
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### Alternatives to calculating the rank of the information matrix in determining if the model is identifiable

I have a known non-linear model $h \in \mathbf{R}^n$: $$y = h(\theta) + \epsilon,$$ where $\theta\in \mathbf{R}^m$ is a parameter vector, and $\epsilon$ is a normal random variable with zero mean ...
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### Different version of Wald test statistic formula

I came across two formulas for the Wald test statistic in a maximum likelihood framework: One has $(R\hat{\theta}-r)'(RI_n^{-1}R')^{-1}(R\hat{\theta}-r)$, where $I_n^{-1}$ is the inverse of the ...
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### Statistics : 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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### Inference for Maximum Likelihood Estimator Using Particle Filter

How does one compute standard errors for the MLE when using a particle filter approximation to the likelihood? I know that the estimator is asymptotically normal and that the variance-covariance ...
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I'm reading this paper on Natural Policy Gradient https://papers.nips.cc/paper/2073-a-natural-policy-gradient.pdf and have some questions regarding how it works. I'm coming at this from an ML ...
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### Fisher information for MLE with constraint

Supposing I have a probability distribution $f(x|\vec\theta)$, where $x$ is a random variable and $\vec\theta$ is a vector of distribution parameters. I also know that parameters $\vec\theta$ should ...
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### Why does Fisher use covariance when only variance is needed?

With reference to the following image from here: (can not inline it due to unsupported format) https://wikimedia.org/api/rest_v1/media/math/render/svg/9af8aa035642689bb2004047416b069a15406447 If we ...
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### Fisher information for models that are not one-to-one

When a parameterized data model and corresponding pdf are known, the Cramér-Rao lower bound provides an lower bound for the variance of an estimator of one of the parameters. That is, given the data ...
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### Observed Fisher Information & Cramer-Rao bound

The Cramer-Rao bound is usually derived for the "expected" or "total" fisher-information. Is a similar result possible for the observed-fisher information?
Background The Fisher information for a linear Gaussian model is $\mathcal{I}_{\theta} = \frac{X X^T}{\sigma^2}$. This is used in optimal experiment design techniques, for example, maximisation of \$|...