Which sufficient statistic is for which parameter

Question

Suppose I have n samples $x_1,x_2,...x_n$ sampled from a $\mathcal{N}(\mu,\sigma^2)$. I need to find the sufficient statistics for $\mu,\sigma^2$. I write the likelihood

$$ \mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=1}^n x_i^2 + \sum_{i=1}^n \mu^2 - 2\mu\sum_{i=1}^n x_i\right)\right) $$

$$\mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{n\mu^2}{2\sigma^2}\right) \exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=1}^n x_i^2 - 2\mu\sum_{i=1}^n x_i\right)\right)$$

Then we say that by using factorization theorem $T_1 = \sum_{i=1}^n x_i^2$ and $T_2 = \sum_{i=1}^nx_i$ are sufficient statistics. I want to ask how we come with that the first sufficient statistic is for $\mu$ and other one is for $\sigma^2$.

I know like MLE is a function of sufficient statistic with that probably we can say that first one is for $\mu$ and other is for $\sigma$(not sure about this thing) but what if I don't know the MLE of some distribution, then how to figure out.

Answer 1

A sufficient statistic should be understood as function of the sample gathering the important information about the parameter (possibly multidimensional). Note that a sufficient statistic could be multidimensional even when you work with unidimensional parameters. For instance, consider an uniform, $U(\theta-1, \theta +1).$ This distribution has just one parameter $\theta$ and its (minimal) sufficient statistic is $(X_{(1)},X_{(n)}).$

That said, it is very common to obtain a sufficient statistic of the form $(T_1,\ldots, T_k)$ when working with a $k-$ multidimensional parameter $(\theta_1,\ldots,\theta_k).$ An easy way to compute a sufficient statistic for one of the parameters, say $\theta_1,$ is fixing as constants the rest of parameters and using factorization theorem. In your case, $\mathcal{N}(\mu,\sigma^2)$, say we want to compute a sufficient statistic for $\mu.$ We can consider $\sigma ^2$ as a known value and the likelihood would be: $$\mathcal{L}=C\cdot \exp \left( - \dfrac{1}{2\sigma ^2} \left( n \mu^2 - 2 \mu \sum_{i=1}^n x_i \right) \right),$$ where $C$ is a constant (does not depend on $\mu$). Now you can easily apply factorization theorem to see that $T_2=\sum_{i=1}^n x_i$ is a sufficient statistic for $\mu.$

Using MLE for detecting which sufficient statistic is good to estimate which parameter is also a valid approach (at least most times), however it just identifies one of the statistics (in the best case).