# Get a Fisher information matrix for linear model with the normal distribution for measurement error?

For given linear model $y = x \beta + \epsilon$, where $\beta$ is a $p$-dimentional column vector, and $\epsilon$ is a measurement error that follows a normal distribution, a FIM is a $p \times p$ positive definite matrix.

How to find elements of the matrix?

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Let $\gamma$ denote the gaussian distribution of $\epsilon$. The likelihood of the model is $$\gamma(y - x \beta)$$ where $y$ is your observation and $\beta$ is the parameter. You can now apply the definition of the Fisher Information matrix, $$I = \text{var} \left( \nabla_\beta \log \gamma(Y - x\beta) \right).$$