On a characterization theorem of symmetry about a point

Abstract

Let X1,…,Xn be i.i.d. copies of random variable X where 0<E|X|<∞ and let X̄=1n∑i=1nXi. One can show that X1−X̄,…,Xn−X̄ are exchangeable, and as a result identically distributed, but not independent. We use this result to prove that for n≥3, X is symmetric about a point if and only if X1−X̄ is symmetric. The theorem is applied for testing symmetry about an unknown point.