Question

Let \(X_1,X_2,X_3\) be a random sample from a distribution having probability mass function
\[ f_{\theta}(x)=\begin{cases}\theta & \text{if } x=1\\ 1-\theta & \text{if } x=2\\ 0, & \text{otherwise},\end{cases} \]
where \(\theta\in\Theta=(0,1)\). Let \(\underline{X}=(X_1,X_2,X_3)\). Then which of the following is NOT a sufficient statistic for \(\theta\)?

• A. \(T_1(\underline{X})=(X_1-X_2,\ X_1+X_2,\ X_1+X_3)\)
• B. \(T_2(\underline{X})=(X_1+X_2,\ X_1-X_3,\ X_2+X_3)\)
• C. \(T_3(\underline{X})=(X_1-X_2,\ X_2-X_3,\ X_3)\)
• D. \(T_4(\underline{X})=(X_1+X_2,\ X_3)\)

Hint:

A statistic is sufficient only if it lets you recover the count of \(1\)s among \(X_1,X_2,X_3\); check whether two samples with different counts can give the same statistic value.

Answer 1

The Correct Option is B

Step 1: Recall the sufficiency rule.
A statistic is sufficient exactly when it determines $n_1$ (count of $1$s), or equivalently the sample sum.

Step 2: Check invertibility of each linear map.
$T_1$: coefficient matrix determinant $=-2\ne0$, invertible, recovers everything. $T_3$: upper triangular, determinant $1$, invertible. $T_4$: sum of components gives $X_1+X_2+X_3$ directly.

Step 3: Check $T_2$.
Coefficient matrix determinant $=0$, singular, information lost.

Step 4: Confirm the lost information matters.
$(1,2,1)$ ($n_1=2$) and $(2,1,2)$ ($n_1=1$) both give $T_2=(3,0,3)$, but different likelihoods, so $T_2$ is not sufficient.

Final Answer:
\[ \boxed{T_2(\underline{X})=(X_1+X_2,\ X_1-X_3,\ X_2+X_3)} \]