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?

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    $\begingroup$ Please add the self-study tag and read the tag wiki. Did you find out what $T$ is here? $\endgroup$ – StubbornAtom Oct 19 at 10:45
  • $\begingroup$ $T = \left( \Sigma_{i=1}^m X_i^{2} + \Sigma_{i=1}^n Y_i^{2} , \Sigma_{i=1}^m X_i \right)$ ? $\endgroup$ – j200932 Oct 19 at 11:08
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    $\begingroup$ 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. $\endgroup$ – StubbornAtom Oct 19 at 15:43

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