Guys can you help me with this? I’ve tried to apply the definition of sufficient statistics but the professor only gave us simple examples so I don’t know how to compute it.
$$X\sim f_θ$$
$$f_θ(x)=\frac{θ}{2}(θx)^2\exp\{−θx\}\text{ for }x>0$$
Moreover, let $X_1,…,X_n$ be a sample from $X$. a) Find a sufficient statistic for the model b) Find the MLE of $θ$ and the MLE of $E(X)=\frac{3}{θ}$
\begin{align} \prod_{i=1}^n f_\theta(x_i) & = \left(\frac \theta 2\right)^n ((\theta x_1) \cdots (\theta x_n))^2 \exp\{−\theta (x_1+\cdots+x_n)\} \\[10pt] & = \Big(\, \overbrace{\theta^{3n} \exp(-\theta(x_1+\cdots + x_n))}^\text{first factor}\,\Big) \cdot \Big(\, \overbrace{ \frac{x_1\cdots x_n}{2^n}}^\text{second factor} \, \Big) \tag 1 \end{align}
Fisher's factorization theorem says that if this can be written as a product of two factors in which
then that specified function, evaluated at the sample, is a sufficient statistic.
Therefore $X_1+\cdots+X_n$ is a sufficient statistic.
If $L(\theta) = $ the expression on line $(1)$ above, then $$ \ell(\theta) = \log L(\theta) = 3n\log\theta - \theta(x_1+\cdots+x_n) + \text{constant} $$ so $$ \ell\,'(\theta) = \frac{3n}\theta - (x_1+\cdots+x_n) \quad \begin{cases} >0 & \text{if } 0 \le \theta < 3/\overline{x}, \\[6pt] <0 & \text{if } \theta>3/\overline{x}. \end{cases} $$ Therefore $\widehat \theta = 3/\overline x.$
By equivariance of MLEs, the MLE of $3/\theta$ is $\overline x.$