# 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?

## 1 Answer

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. – Xi'an Nov 10 '14 at 19:31