Let $X_1,X_2,X_3,....X_n$ be a random sample from a distribution with the probability density function
$$f(x|\theta) = \begin{cases} \dfrac{x}{\theta^2}e^{\frac{-x}{\theta}}, & \text{if $x>0$; $\theta>0$} \\ 0, & \text{otherwise} \end{cases}$$
Which of the following statistics is(are) sufficient but NOT complete?
$A=\bar X$
$B=\bar X^2+3$
$C=( X_1,\sum_{i=2}^{n}X_i) $
$D=(X_1,\bar X)$
My input
The joint probability density function of the sample is
$f(x_1,x_3,x_2...x_n|\theta) =\dfrac{\prod_{i=1}^{n}x_i}{\theta^{2n}}\bigg(e^{\dfrac{-\sum_{i=1}^{n}x_i}{\theta}}\bigg)$
Let $T(x)=\sum_{i=1}^{n}x_i$ .We see that the pdf can be expressed as a function that depends on the sample only through $T(x)$. Thus $T(x)$ is a sufficient statistic for $\theta$
$A$= is sufficient statistic because $\bar X$ is a function of $\sum_{i=1}^{n}x_i$ by dividing the whole sum by $n$ gives $\bar X$ ( I am not sure if this reasoning of mine is valid I just can't explain it to someone in brief. I need a perfect reasoning for that so someone tell me that.)
$B$= is a sufficient statistic because any one to one function of sufficient statistic T is also sufficient statistic. For example, if $x>0$ $x^2$ will be a one to one function.
$C$= Sum of these two statistics give $\sum_{i=1}^{n}x_i$
$D$= I am not sure how to calculate sufficient statistic in this one because I am not able to derive $\sum_{i=1}^{n}x_i$ with any linear combination. So my intuition says it's not sufficient statistics.
Completeness is the next step but I am stuck in calculating sufficient statistic. I am learning these concepts so please give me any tips ,advice before calculating these things.