By taking X to be a two-dimensional standardized random variable with zero mean and identity covariance matrix, interpret Q4(W), defined in the previous exercise, as a directional standardized kurtosis. Taking ϵ1 as defined in Exercises 2.36 and 2.37, show, using the polar representation, that that ϵ2 − ϵ1 and ϵ4 − ϵ1 are both invariant under rotation of X, but change sign under reflection. Find expressions for these semi-invariants in terms of the κs. Interpret τ0 as the mean directional kurtosis and express this as a function of 4. Discuss the statistical implications of the following conditions:

16τ

2 1 = 9(κ30 + κ12)

2 + 9(κ03 + κ21)

2 16τ

2 3 = (κ30 − 3κ12)

2 + (κ03 − 3κ21)

2

(i) τ0 = 0;

(ii) τ2 = 0;

(iii) τ4 = 0;

(iv) ϵ4 − ϵ2 = 0.