Maximum sufficient statistics?

Question

Suppose $\{X_i: i = 1, \ldots, n\}$ follows the parametric family of distribution $f(x_1,\ldots,x_n|\theta)$. According to Neyman-Fisher Factorization Theorem, if a statistic $T(X_1,\ldots,X_n)$ is such that $f$ can be factorized in the form $f(x_1,\ldots,x_n|\theta)=g\left(\theta, T(x_1,\ldots,x_n)\right) h(x_1,\ldots,x_n)$, then $T$ is a sufficient statistic for the family. Then, is the identity function $T(X_1, \ldots, X_n)=(X_1, \ldots, X_n)$ a sufficient statistic?

Answer 1

The identity statistic is always sufficient for all parameters in the family, because sufficiency relates to data reduction. Tautologically, all the information about the parameters that is contained in the sample, remains in the the sample--that is to say, because no data reduction is achieved, no information is lost.

The notion that a statistic must be a function whose image is a subset of $\mathbb R$ is incorrect. Statistics need not map to a real number. A statistic is a function of a sample that does not depend on any unknown parameters. Last time I checked, functions are not restricted to having $\mathbb R$ as the codomain. Indeed, if you were to impose such a restriction, then the concept of a sufficient statistic, not to mention minimal sufficient statistics, would not be defined for many parametric distributions, and one would also wonder what we would call realizations of a multinomial random variable or even a bivariate normal random variable.