# Compute the information matries related to normal distribution

This is a problem that I have trouble with.

Suppose that we have $$X_{1}, \ldots, X_{m}$$ are iid $$N\left(\mu, \sigma^{2}\right), Y_{1}, \ldots, Y_{n}$$ are iid $$N\left(0, \sigma^{2}\right),$$ the $$X$$ 's are independent of the $$Y$$ 's where $$-\infty<\mu<\infty, 0<\sigma<\infty$$ are the unknown parameters. Suppose that $$\mathbf{T}=\mathbf{T}(\mathbf{X},$$ $$\mathbf{Y})$$ is the minimal sufficient statistic for $$\theta=\left(\mu, \sigma^{2}\right) .$$ Evaluate the expressions of the information matrices $$I_{\mathrm{XY}}(\theta)$$ and $$I_{\mathrm{T}}(\theta),$$ and then compare these two information contents.

The information matrix of $$I_{\mathrm{XY}}(\theta)$$ is not hard to compute, but I'm not sure how to deal with $$I_{\mathrm{T}}(\theta)$$. Also, the problem asks me to compare these two matrices, so I wonder whether there is relation between the information matrix of the original data and that of a sufficient statistic ( in my book the author mentions the one-parameter case, so I'm asking about the case when $$\theta$$ is a vector). Can anyone help?

• Please add the self-study tag and read the tag wiki. Did you find out what $T$ is here? – StubbornAtom Oct 19 at 10:45
• $T = \left( \Sigma_{i=1}^m X_i^{2} + \Sigma_{i=1}^n Y_i^{2} , \Sigma_{i=1}^m X_i \right)$ ? – j200932 Oct 19 at 11:08
• Yes, a minimal sufficient statistic for $\theta$ is $T=(\sum X_i,\sum X_i^2+\sum Y_i^2)$. Because this is sufficient, you would expect both $T$ and the original data $(X,Y)$ to have the same information about $\theta$, i.e. the information matrices should be the same. – StubbornAtom Oct 19 at 15:43