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. $274.$