# Minimal Sufficient Statistic for Gaussians with different means

I have the following problems on my Statistics course (using Casella and Berger's book) problem set:

1) Let $Y_{i} = X_{i}'\theta + U_{i}$ where $\theta \in \mathbb{R}^k$ and $U_{i}$ are iid $N(0, \sigma^2)$ random variables and $X_{i}$ is a fixed vector for each $i$. Find the minimal sufficient statistic when $\sigma^2$ is known.

2) Find the minimal sufficient statistic when $\sigma^2$ is unknown.

I was able to represent this model in a vectorial way, write down the vectorial joint density for $Y$ and show that the OLS regressor of $Y$ on $X$ is a sufficient statistic for $\theta$. However, I'm having trouble with showing that it's the minimal one and also with the case when $\sigma^2$ is unknown. Any ideas how to procede?

• One thing that might help is that the minimum dimension of a sufficient statistic is the dimension of the parameter vector, see stats.stackexchange.com/questions/159101/…. So if you can show that the dimension of the OLS regressor equals the dimension of $\theta$, and that the OLS regressor is a sufficient statistic (which you have done), then you can conclude that the OLS regressor is minimal sufficient. – jbowman Jun 9 '18 at 20:15
• You should add the [self-study] tag, please! – kjetil b halvorsen Jun 9 '18 at 20:16
• @jbowman I wasn't aware of this result. This helps a lot. – Raul Guarini Jun 9 '18 at 20:37