# Equivalent definitions of a sufficient statistic

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?

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.

• sorry but I fear this explanation does not make sense as it confuses densities and events. Please check any reference textbook like Lehmann and Casella or Casella and Berger. Nov 10, 2014 at 19:31