By taking X to be a two-dimensional standardized random variable with zero mean and identity covariance matrix, interpret Q3(w), defined in the previous exercise, as a directional standardized skewness. [Take w to be a unit vector.] Show that, in the polar representation, ϵ3 − ϵ1 is invariant under rotation of X, but changes sign under reflection.

Find an expression for this semi-invariant in terms of κ30, κ03,κ21 and κ12. Discuss the statistical implications of the following conditions:

(i) 4ρ

2 23 − 3ρ

2 13 = 0;

(ii) ρ

2 13 = 0;

(iii) ϵ3 − ϵ1 = 0.