On a characterization theorem of symmetry about a point

摘要

Let _(X1,., X n) be i.i.d. copies of random variable X where 0 < E|X| < ∞ and let X?=1n∑i=1nXi. One can show that X1-X?,. .,Xn-X? 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-X? is symmetric. The theorem is applied for testing symmetry about an unknown point.