= Solution
The joint density depends on the sample through
$$
\sum_{i=1}^nX_i^2,
$$
not merely through $\sum_i|X_i|$. For example, after padding with zeros when $n>2$, take
$$
x=(1,1,0,\ldots,0),
\qquad
y=(2,0,0,\ldots,0).
$$
Both samples have sum of absolute values $2$, but
$$
\frac{p_\sigma(x)}{p_\sigma(y)}
=\exp\left(\frac1{\sigma^2}\right),
$$
which depends on $\sigma$. The likelihood-ratio necessary condition for sufficiency fails, so
$$
\boxed{\sum_{i=1}^n|X_i|\text{ is not sufficient}.}
$$
Back to article page
