Equivalent definitions of a sufficient statistic

Question

I'm trying to understand the definition of a sufficient statistic for continuous random variables given in Introduction to Mathematical Statistics by Hogg and Craig (7th edition). Let $X_1,X_2,...,X_n$ be a random sample with joint pdf $f(x_1,x_2,...,x_n;\theta)$, $\theta \in \Omega$, and $T(X_1,...,X_n)$ be a statistic with pdf $f_T(y;\theta)$. Wikipedia, among other sources, defines $T$ to be sufficient for $\theta$ if and only if the conditional distribution of $X_1,X_2,...,X_n$, given $T=t$, does not depend on $\theta$. On the other hand, Hogg and Craig defines $T$ to be sufficient for $\theta$ if and only if $$\frac{f(x_1,x_2,...,x_n;\theta)}{f_T(T(x_1,...,x_n);\theta)}$$ does not depend on $\theta$.

Here's my question: are the two definitions equivalent, and if so, how does one prove this?

Answer 1

As Stéphane Laurent aptly pointed out, this is nothing but Fisher-Neyman factorization theorem.

That is, succinctly, if $\mathbf X\sim f(\mathbf x|\theta), ~T(\mathbf X) ~\sim f_T(T(\mathbf x) |\theta),~T$ being sufficient for $\theta, $ then $f(\mathbf x|\theta)/f_T(T(\mathbf x) |\theta)$ is constant as function of $\theta$ for every value of $\mathbf x. $

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Reference:

$\rm [I]$ Statistical Inference, George Casella, Roger L. Berger, Wadsworth, $2002, $ sec. $6.2, $ p. $274.$

Answer 2

Yes. Both are the same. According to the 1st definition $f(x_1,x_2,...,x_n |T=t)=$ independent of θ.

So, LHS $=f(x_1, x_2,....,x_n , T=t)/f(T=t)$. By definition: $P(A|B)=P(A ∩ B)/P(B)$.

1. If $x_1,x_2,....,x_n$ are such that $T(x_1,...,x_n)=t$ then we can write $\{x_1,...,x_n\} \rightarrow \{T=t\}$. I.e. $\{x_1,....,x_n\} ⊂ \{T=t\}$.

   I.e. $\{x_1,....,x_n\}∩\{T=t\} = \{x_1,...,x_n\}$.
   Then $f(x_1, x_2,....,x_n , T=t) = f(x_1,x_2,....,x_n)$.
   Therefore LHS = $f(x_1,x_2,....,x_n)/f(T=t)$

2. If $x_1,....,x_n$ are such that $T(x_1,...,x_n) ≠ t$ then $\{x_1,....,x_n\} ∩ \{T=t\} = Ø$

   So, $f(x_1, x_2,....,x_n , T=t) = 0$. Therefore LHS = 0 (always independent of θ). So we only have to concentrate on the previous case to show $f(x_1,x_2,....,x_n)/f(T=t)$ is independent of θ. This is precisely the 2nd definition.