# Cramer-Rao lower bound for the variance of unbiased estimators of $\theta = \frac{\mu}{\sigma}$

Let $$X_1, \cdots, X_n$$ be a sample from the $$N(\mu, \sigma^2)$$ density, where $$\mu, \sigma^2$$ are unknown. I want to find a lower bound $$L_n$$ which is valid for all sample-sizes $$n$$ for the variance of unbiased estimators of $$\theta = \mu/\sigma$$.

I know that for one-dimensional parameters the Cramer-Rao lower bound for unbiased estimators is given by $$\frac{1}{I(\theta)}$$, where $$I(\theta)$$ is the fisher information. Since $$\mu, \sigma^2$$ are unknown the Fisher information becomes a $$2\times 2$$ matrix. So how can the Fisher information matrix help me to find the Cramer-Rao lower bound?

Any hints will be appreciated.

Suppose $$\theta=(\mu,\sigma^2) \in \Omega$$ is the two-dimensional parameter where $$\Omega=\mathbb R\times (0,\infty)$$.
You are interested in the parametric function $$g:\Omega \to \mathbb R$$ where $$g(\theta)=\frac{\mu}{\sigma}$$.
Then $$g$$ is differentiable and its gradient at $$\theta$$ is denoted by $$\nabla g(\theta)$$.
The Cramér-Rao lower bound (see this or this) for the variance of any unbiased estimator of $$g(\theta)$$ is the inverse of Fisher information about $$g(\theta)$$, which is given by
$$\text{CRLB}(g(\theta))=\nabla g(\theta)^T(I(\theta))^{-1}\nabla g(\theta)$$