Sufficient Statistics, Factorization Theorem, Exponential Family

Sufficient Statistics
What's Statistic? 
- Let $X_{1},..., X_{n}$ ~ $F_{\theta}$ (some function), and let S be a function of {$X_{1},..., X_{n}$ }. S(X) is a statistic if it does NOT depend on any unknown quantities including $\theta$, which means you can actually compute S(X). 
-Statistic examples are sample mean, min, max, median, order statistics... etc. So even if you don't know what the $\theta$ is you can compute those. 

What's Sufficient Statistic?
- Let $X_{1},..., X_{n}$ ~ $F_{\theta}$ (some function), and let S be a function of {$X_{1},..., X_{n}$ }. T(X) is called a sufficient statistic for $\theta$ if it is a statistic and the conditional probability P(X|T) does NOT depend on $\theta$. 

Example 1 
Let $X_{1},...,X{n}$ be a random sample from a distribution with the following density function. $f(x|\theta)= \frac {2x}{\theta}\exp (\frac{-x^2}{\theta})$, x>0 Show that $\sum_{i=1}^{n}X_{i}^2$ is a sufficient statistic for $\theta$
Solution??!!

   
Factorization Theorem
-Suppose $X_{1},..., X_{n}$ ~ $F_{\theta}$, then a if and only if condition for T=T( $X_{1},..., X_{n}$) to be a sufficient statistic for $\theta$ is that the joint probability factor in the form
f(X|$\theta$)=g[T,$\theta$]$\cdot$h(X).


The Exponential Family
Suppose $X\sim f_{\theta}$  where $\theta$ is a vector parameter with k components $\theta_{1},..., \theta_{k}$ and and $f_{\theta}$ is a probability density. 

$f_{\theta}(X)= \exp {[\sum_{i=1}^{k}]C_{i}(\theta)\cdot T_{j}(X)}+d(\theta)+ s(X)$  $\Leftrightarrow$ 

we say $f_{\theta}$ belongs to the exponential family of distributions.