When introducing the notation of a sufficient statistic, Hogg uses a peculiar sort of notation at the bottom of page 381 of the 7th edition textbook: given the statistic $$Y_1 = u_1(X_1,X_2,...,X_n)$$ and $$(X_1,X_2,...,X_n)∈\{(x_1,x_2,...,x_n):u_1(x_1,x_2,...,x_n)=y_1\},$$ this sufficient statistic partitions the sample space such that the conditional probability of $X_1,X_2,...,X_n$ does not depend on $\theta$. Here, it is implicit that $X_1,...,X_n$ constitutes an iid random sample and that $\theta$ represents the (single-variate) parameter of the shared distribution on each $X_i$.

My question here is on what exactly Hogg means by $$(X_1,X_2,...,X_n)∈\{(x_1,x_2,...,x_n):u_1(x_1,x_2,...,x_n)=y_1\}.$$ It is clear to me that it reads as "the set $(X_1,...,X_n)$ is a subset of all such sets of $(x_1,...,x_n)$ such that the mapping $u_1$ takes $x_1,...,x_n$ into $y_1$", but to be consistent with the prior convention adopted when defining random samples (top of page 204), it must be that $x_1,...,x_n$ represent the realizations of the random sample.

If my understanding is correct, the fact that the sufficient statistic partitions the sample space indicates that the sample space $\Omega$ can be divided into the two mutually exclusive and exhaustive subsets $$\omega = \{(X_1,...,X_n):u_1(X_1,...,X_n)=Y_1\}$$ and $$\omega_0 = \{(X_1,...,X_n):u_1(X_1,...,X_n) \neq Y_1\};$$ this does not require any such reference to the realizations of the random sample whatsoever.

So why then is it important to note that our random sample $(X_1,...X_n)$ is a subset of the restriction by the mapping of $u_1$ applied to all possible values in $\Bbb R^n$ into the value $y_1$? When considering the partitioning of the sample space by this statistic, all that is important to defining a sufficient statistic is the function of the random sample as defined by $Y_1$, so I don't understand the purpose of Hogg needing to make use of realizations with respect to the function $u_1$ at all.

Perhaps it is just that my understanding of realizations with respect to their use in statistical definition notation is not quite sufficient? For reference, my background in understanding statistics is upon a statistics-based foundation; this is to be contrasted with learning statistics with a mathematics-based foundation (measure theory). However, my understanding and ability with pure mathematics covers all of the fundamentals necessary to approach statistics from a measure theoretical perspective, so if there is anything about realizations that is relevant here requires such a more advanced perspective to be taken, feel free to let me know.


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