1. While d_22 implies d_21, as the covariance cannot be different from zero if the variance is zero, is it clearer to write H_0 : d_22 = d_21 = 0, as this makes it more explicit that we are testing two parameters.
2. Of these two parameters,
- d_22 is on the boundary of the parameter space under the null and only testing H_0: d_22 =0 would result in a mixture distribution of a point mass at zero and a chi-square distribution with one degree of freedom as explained on the slides for d_11.
- d_21 is not on the boundary of the parameter space under the null and just testing this parameter would result in a chi-square distribution with one degree of freedom as usual.