The definition of sufficient statistic is as follows:
A statistic $T(X_1,...,X_n)$ is sufficient for parameter $\theta$ if the conditional distribution of $X_1,...,X_n$, given that $T=t$, does not depend on $\theta$ for any value of $t$.
I often see in resources that in the case we have iid $X_1,...,X_n \sim N(\mu, \sigma^2)$, $\bar{X}$ is a sufficient statistic for $\mu$ when $\sigma^2$ is known. $\sigma^2$ has to be known otherwise the Factorization theorem doesn't factorize properly.
However, regardless of whether $\sigma^2$ is known or unknown, the conditional distribution of $f(X_1,...,X_n; \bar{X}, \mu, \sigma^2)=f(X_1,...,X_n|\bar{X}, \sigma^2)$ (from multivariate normal theory). Even when $\sigma^2$ is unknown, the conditional distribution of the data $\textbf{X}$ given $\bar{X}$ is independent of $\mu$, suggesting that $\bar{X}$ is a sufficient statistic for $\mu$ .
In general, it seems that if I have density defined by a parameter vector $\theta^\top=(\theta_1^\top, \theta_2^\top)$, where $T_1$ is a sufficient statistic for $\theta_1$ when $\theta_2$ is known, then regardless of whether $\theta_2$ is known or unknown, $f(X_1,...,X_n; T_1, \theta_1, \theta_2)=f(X_1,...,X_n;T_1, \theta_2)$(Essential Statistical Inference pg. 59 Boos and Stefanski).
- Why does the Factorization Theorem fail to identify $\bar{X}$ as a sufficient statistic when $\sigma^2$ is unknown, even though it satisfies the sufficient statistic definition?
- How necessary is it to declare that a statistic is sufficient only when certain nuisance parameters are known, if the conditional distribution is still independent of the parameter of interest?