If X and (−1)X both have the same distribution function F, defined on Rp, then F is said to be diagonally symmetric about 0. Show that if F is diagonally symmetric about 0, then for every ℓ ∈ Rp, ℓ ⊤X...

If X and (−1)X both have the same distribution function F, defined on Rp, then F is said to be diagonally symmetric about 0. Show that if F is diagonally symmetric about 0, then for every ℓ ∈ Rp, ℓ ⊤X has a distribution function symmetric about 0. This property is related to the multivariate median unbiasedness: If for every ℓ ∈ Rp, ℓ⊤(Tn − θ) has median 0, then Tn is multivariate median unbiased for θ. Verify (3.17). [Sen 1990]